1 Introduction↩︎

The investigations reported here had their origins in work [1] on the Keiper-Li criterion for the Riemann hypothesis [2], [3]. The Keiper-Li criterion involves positive valued coefficients \(a_n\) arising in expansions of the Riemann zeta function. The new representation of the \(a_n\) reported in [1] involved a combination of two sets of coefficients \(C_{n,p}\) and \(\Sigma^{\xi}_p\), again positive valued. This representation enabled the accurate calculation of the first 4000 coefficients \(a_n\). The coefficients \(C_{n,p}\) obeyed a recurrence relation, and had representations involving the classical Euler sums. A deeper understanding of the asymptotic behaviour of the \(C_{n,p}\) as the two integer parameters \(n\) and \(p\) tended to infinity was sought, and naturally involved results from the extensive literature on Euler sums [4]-[5].

In this paper we address Euler sums of the form \[\begin{align} M(m,n_0,n_1,n_2, \ldots, n_t) &= \sum_{k=1}^\infty \frac{H(k)^m}{k^{n_0} (k+1)^{n_1} (k+2)^{n_2} \cdots (k+t)^{n_t}}, \end{align}\] for nonnegative integers \(m\) and \((n_i)\), with \(m \geq 1\) and \(n_0 + n_1 + \cdots + n_t \geq 2\), where \(H(k) = H_k = \sum_{j=1}^k 1/j\) is the harmonic function (we use both notations interchangeably below). However, the techniques presented below are applicable to Euler sums of even more general forms. We focus on Euler sums having a common order \(r\), where \(r = m + n_0 + n_1 + \cdots + n_t\). We combine results from the literature with many new results, in an effort to say as much as possible about systems of order \(r\) ranging from 3 to 12. The complexity of these analyses increases rapidly with \(r\).

Among the most striking results of this paper are the linkages between Euler sums and Stieltjes constants. The latter can be defined by: \[\lim_{n\rightarrow \infty}\left[\sum_{k=1}^n \frac{(\log k)^p}{k}-\frac{(\log n)^{p+1}}{p+1}\right]=\gamma_p. \label{intro1}\tag{1}\] The Stieltjes constants \(\gamma_p\) have a sign which varies in a complicated way as \(p\) increases and their modulus increases. We define their equivalent for Euler sums as \[\lim_{n\rightarrow \infty}\left[\sum_{k=1}^n \frac{H_k^{p-1}}{k}-\frac{1}{p} H_n^p\right]=\gamma_p^H. \label{intro2}\tag{2}\] Here the harmonic Stieltjes constants \(\gamma_p^H\) are all positive, and again increase rapidly as \(p\) increases. The harmonic Stieltjes constants are shown to be given by a sum over a set of primitive sums which form a basis for the Euler sums of order \(p\).

2 Previous results on Euler Sums↩︎

We now present selected results from the literature on Euler sums, including relatively recent results due to the late Jonathan Borwein and collaborators [6], [7], which were obtained using the techniques of experimental mathematics to complement analysis. Two of the sets of sums they consider are: \[\begin{align} s_h(m,n) &= \sum_{k=1}^\infty \frac{H_k^m}{(k+1)^n} \tag{3} \\ \sigma_h (m,n) &= \sum_{k=1}^\infty \frac{H_k^{(m)}}{(k+1)^n}, \tag{4} \end{align}\] where \(H_k^{(m)} = \sum_{j=1}^k 1/j^m\). We define slight modifications of these: \[\begin{align} {\cal I}_h(m,n) &= \sum_{k=1}^\infty \frac{H_k^m}{k^n},{\cal J}_h(m,n) = \sum_{k=1}^\infty \frac{H_k^{(m)}}{k^n}. \label{mydefs1} \end{align}\tag{5}\] Then [6]: \[\begin{align} {\cal J}_h(m,n) &= \sum_{k=1}^\infty \frac{H_k^{(m)}}{k^n}=\sigma_h (m,n)+\zeta (m+n), \label{EBBG3} \end{align}\tag{6}\] where \(\zeta(p) = \sum_{k \geq 1} 1/k^p\) is the Riemann zeta function. For \(m=1\), this is \[\begin{align} {\cal I}_h(1,n)-s_h(1,n) &= \zeta (n+1). \label{myeqn} \end{align}\tag{7}\] For the special case \(m=2\) under particular investigation in [6], [7], \[\begin{align} {\cal I}_h(2,n)-s_h(2,n) &= 2 {\cal I}_h(1,n+1)-\zeta (n+2)=2 s_h(1,n+1)+\zeta (n+2). \label{EBBG4} \end{align}\tag{8}\] For both these cases, the right-hand side tends down to unity as \(n\rightarrow \infty\).

The sums \({\cal I}_h(n,p)\) can be represented in terms of the \(s_h(m,n)\) as follows: \[\begin{align} {\cal I}_h(n,p) &= \sum_{k=1}^\infty \frac{H_k^n}{k^p}=\zeta (n+p)+\sum_{m=1}^{n} \binom{n}{m} s_h(m,p+n-m). \label{EBBG4a} \end{align}\tag{9}\] We can define the order of this expression to be \(n+p\), i.e., the sum of the powers of \(H_k\) and \(k\) on the left-hand side. The sum of the arguments of \(s_h\) on the right-hand side is also \(n+p\). The dual expression to equation (9 ) is \[\begin{align} s_h(n,p) &= \sum_{k=1}^\infty \frac{H_k^n}{(k+1)^p}=(-1)^n\zeta (n+p)+\sum_{m=1}^{n} \binom{n}{m} (-1)^{m-n} {\cal I}_h (m,p+n-m). \label{EBBG4aa} \end{align}\tag{10}\] Euler provided the solution for \(\sigma_h (1,m)=s_h(1,m)\) and thus for \({\cal J}_h(1,m)={\cal I}_h(1,m)\) for all \(m\ge 2\): \[\begin{align} \sigma_h (1,m) &= \frac{m}{2} \zeta (m+1)-\frac{1}{2} \sum_{k=1}^{m-2} \zeta (m-k) \zeta (k+1). \label{EBBG5} \end{align}\tag{11}\] Another useful relationship is the reflection formula, valid for \(m,n\ge 2\): \[\begin{align} \sigma_h (m,n)+\sigma_h (n,m) &= \zeta(m)\zeta(n)-\zeta(m+n), \label{EBBG6} \end{align}\tag{12}\] or, written with different notation, \[\begin{align} {\cal J}_h(m,n)+{\cal J}_h(n,m) &= \zeta(m)\zeta(n)+\zeta(m+n), \label{EBBG7} \end{align}\tag{13}\] so that \(2{\cal J}_h(m,m)=\zeta(m)^2+\zeta(2m)\) for \(m\ge 2\).

Euler [7] was able to derive the following expansions in terms of zeta functions, for the particular case where the sum of parameters \(s+t\) is odd, and \(t>1\). The first is for \(s\) odd, \(t\) even: \[\begin{align} \sigma_h(s,t) &= \frac{1}{2} \left[ \binom{s+t}{s}-1\right]\zeta(s+t)+\zeta(s) \zeta(t) \nonumber \\ &-\sum_{j=2}^{(s+t-1)/2} \left[\binom{2 j-2}{s-1}+ \binom{2 j-2}{t-1}\right] \zeta(2 j-1) \zeta(s+t-2 j+1). \label{EBBG8} \end{align}\tag{14}\] For \(s\) even, \(t\) odd: \[\begin{align} \sigma_h(s,t) &= -\frac{1}{2} \left[ \binom{s+t}{s}+1\right]\zeta(s+t) \nonumber \\ &+\sum_{j=2}^{(s+t-1)/2} \left[\binom{2 j-2}{s-1}+ \binom{2 j-2}{t-1}\right] \zeta(2 j-1) \zeta(s+t-2 j+1). \label{EBBG9} \end{align}\tag{15}\] A valuable result derived in [7] is: \[\begin{align} s_h(2,2 n-1) &= \frac{1}{6} (2 n^2-7 n-3) \zeta (2 n+1)+\zeta (2) \zeta(2 n-1)-\frac{1}{2} \sum_{k=1}^{n-2} (2 k-1) \zeta (2 n-1-2 k) \zeta (2 k+2) \nonumber \\ &+\frac{1}{3}\sum_{k=1}^{n-2} \zeta (2 k+1) \sum_{j=1}^{n-2-k} \zeta(2 j+1) \zeta ( k+1-j) \zeta (2n -1-2 k-2 j) . \label{sh2n} \end{align}\tag{16}\]

Table 5 of [6] gives a list of sums \(s_h\) for which the authors were unable to find representations in terms of zeta functions or zeta functions complemented by powers of logarithms of integers and polylogarithms of argument \(1/2\), using various search algorithms. These results highlight the difficulty of finding closed form representations of all the Euler-type sums arising in treatments of the sums \({\cal C}_{n,p}\) for \(p\) large.

The literature on Euler sums [8]-[5] concentrates on the sums \(s_h(m,n)\), \(\sigma_h(m,n)\), \({\cal I}_h(m,n)\) and \({\cal J}_h(m,n)\). Below we will analyze the mixed sums \(M(m,n,p,q)\), which include the existing results for \(s_h(m,n)\) (setting \(n = q = 0\)) and \({\cal I}_h(m,n)\) (setting \(p = q = 0\)) as special cases. The results we present in Appendix 2 include literature results up to order 7, extend them to orders 8–11 and also include selected results for order 12, as well as adding many extra results of all orders (inter alia for \(p, q \neq 0\)).

Note that the extension to orders larger than 7 is not straightforward. Order 8 was painstakingly investigated by Bailey, Borwein and Girgensohn [6], and only the single analytic result for \({\cal I}_h(1,7)\) was found. By refining and extending the numerical methods used, and regarding \({\cal I}_h(2,6)\) as a known quantity, we have been able to obtain all the other sums for order eight in closed form, as reported in Appendix 2. Similar methods have been applied for orders 9, 10, 11 and 12, with the addition of the sums \({\cal I}_h(2,6), \, {\cal I}_h(2,8), \, {\cal I}_h(3,8), \, {\cal I}_h (2,10), \, {\cal I}_h (4,8)\) to the set of assumed constants. In Apppendix 1, the numerical values of these five assumed constants are given to 400 figures accuracy.

Some evaluations of \({\cal I}(m,n)\) are now presented [4]–[8], arranged according to the order \(m+n\) (in the remainder of this section we will drop the \(h\) subscript on \({\cal I}_h\)). For order 3, there is only one: \[\begin{align} {\cal I}(1,2) &= \sum_{k=1}^\infty \frac{H(k)}{k^{2}} = 2\zeta(3). \label{pow3} \end{align}\tag{17}\] For order 4, there are two: \[\begin{align} {\cal I}(1,3) &= \sum_{k=1}^\infty \frac{H(k)}{k^{3}} = \frac{ 1}{4}\left( 5\zeta(4)\right),{\cal I}(2,2) = \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}} = \frac{ 1}{4}\left( 17\zeta(4)\right). \label{pow4} \end{align}\tag{18}\] For order 5, there are three: \[\begin{align} {\cal I}(1,4) &= \sum_{k=1}^\infty \frac{H(k)}{k^{4}} = 3\zeta(5) -\zeta(2)\zeta(3),{\cal I}(2,3) = \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}} = \frac{ 1}{2}\left( 7\zeta(5) -2\zeta(2)\zeta(3)\right), \nonumber \\ {\cal I}(3,2) &= \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}} = 10\zeta(5) +\zeta(2)\zeta(3) . \label{pow5} \end{align}\tag{19}\] For order 6, there are four: \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{5}} &= \frac{1}{4}\left( 7\zeta(6) -2\zeta(3)^2\right),\sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}} = \frac{1}{24}\left( 97\zeta(6) -48\zeta(3)^2\right), \nonumber \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}} &= \frac{ 1}{16}\left( 93\zeta(6) -40\zeta(3)^2\right),\sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}} = \frac{1}{24}\left( 979\zeta(6) +72\zeta(3)^2\right). \label{pow6} \end{align}\tag{20}\] For order 7, there are five: \[\begin{align} {\cal I}(1,6) &= \sum_{k=1}^\infty \frac{H(k)}{k^6} =-\zeta (4) \zeta (3)-\zeta(2) \zeta (5) +4 \zeta (7), \nonumber \\ {\cal I}(2,5) &= \sum_{k=1}^\infty \frac{H(k)^2}{k^5} =-\frac{5}{2}\zeta (4) \zeta (3)-\zeta(2) \zeta (5) +6 \zeta (7), \nonumber \\ {\cal I}(3,4) &= \sum_{k=1}^\infty \frac{H(k)^3}{k^4} = \frac{693}{48} \zeta (7)+2 \zeta (5) \zeta (2)-\frac{51}{4} \zeta (4) \zeta (3), \nonumber \\ {\cal I}(4,3) &= \sum_{k=1}^\infty \frac{H(k)^4}{k^3} =\frac{185}{8} \zeta (7)+5 \zeta (5) \zeta (2)-\frac{43}{2} \zeta (4) \zeta (3), \nonumber \\ {\cal I}(5,2) &= \sum_{k=1}^\infty \frac{H(k)^5}{k^2} =\frac{2051}{16} \zeta (7)+\frac{57}{2} \zeta (5) \zeta (2)+ 33\zeta (4) \zeta (3). \label{pow7} \end{align}\tag{21}\]

For order eight, there is a paucity of results in the literature. A careful study of this case was given by Bailey, Borwein and Girgensohn [6]. It employed an Euler-Maclaurin scheme for the high-precision evaluation of these sums, an enhanced version of which we describe in Section 5 below. The only analytic formula we can give in complete form is one studied by Euler: \[{\cal I}(1,7)=\sum_{k=1}^\infty \frac{H(k)}{k^7} =\frac{9}{2} \zeta (8)-\zeta (6) \zeta (2)-\zeta (5) \zeta (3) -\frac{1}{2} \zeta (4)^2 , \label{pow8a}\tag{22}\] which can be simplified using the analytic expressions for \(\zeta (2 n)\) to \[\begin{align} {\cal I}(1,7) &= \frac{1}{4}\left( 9\zeta(8) -4\zeta(3)\zeta(5)\right). \label{pow8b} \end{align}\tag{23}\]

We have been able to establish solutions for four additional \({\cal I}\) constants if we express them in terms of the set of constants \(\zeta (8)\), \(\zeta(3) \zeta (5)\) and \(\zeta (2) \zeta (3)^2\), together with \({\cal I}(2,6)\): \[\begin{align} {\cal I}(3,5) &= \frac{1}{96}\left( 595\zeta(8) +120\zeta(2)\zeta(3)^2 -576\zeta(3)\zeta(5) -264 {\cal I}(2,6)\right) \tag{24} \\ {\cal I}(4,4) &= \frac{ 1}{144}\left( -14833\zeta(8) -4032\zeta(2)\zeta(3)^2 +16704\zeta(3)\zeta(5) +3744 {\cal I}(2,6)\right) \tag{25} \\ {\cal I}(5,3) &= \frac{1}{288}\left( 67811\zeta(8) +19080\zeta(2)\zeta(3)^2 -78768\zeta(3)\zeta(5) -16920 { \cal I}(2,6)\right) \tag{26} \\ {\cal I}(6,2) &= \frac{ 1}{8}\left( 5843\zeta(8) -328\zeta(2)\zeta(3)^2 +3896\zeta(3)\zeta(5) +456 {\cal I}(2,6)\right) \tag{27} \end{align}\]

An earlier study [6] gives the expansions for all \(s_h(m,n)\) with \(m+n=9\), apart from those coming from (11 ) and (16 ). The basis of function values needed is \(\zeta (9)\), \(\zeta(2) \zeta (7)\), \(\zeta(3) \zeta (6)\), \(\zeta (4) \zeta (5)\) and \(\zeta (3)^3\), the last coming from the double sum in (16 ). These may be used with (9 ) to produce the following evaluations of \({\cal I}(m,n)\) for order \(m+n = 9\): \[\begin{align} {\cal I}(1,8) &= 5\zeta(9) -\zeta(3)\zeta(6) -\zeta(4)\zeta(5) -\zeta(2)\zeta(7) \\ {\cal I}(2,7) &= \frac{ 1}{6}\left( 55\zeta(9) -21\zeta(3)\zeta(6) -15\zeta(4)\zeta(5) -6\zeta(2)\zeta(7) +2\zeta(3)^3\right) \\ {\cal I}(3,6) &= \frac{1}{24}\left( 521\zeta(9) -291\zeta(3)\zeta(6) -306\zeta(4)\zeta(5) +72\zeta(2)\zeta(7) +48\zeta(3)^3\right) \\ {\cal I}(4,5) &= \frac{1}{12}\left( 436\zeta(9) -279\zeta(3)\zeta(6) -258\zeta(4)\zeta(5) +84\zeta(2)\zeta(7) +40\zeta(3)^3\right) \\ {\cal I}(5,4) &= \frac{1}{72}\left( 9442\zeta(9) -14685\zeta(3)\zeta(6) +4752\zeta(4)\zeta(5) +2385\zeta(2)\zeta(7) -360\zeta(3)^3\right) \\ {\cal I}(6,3) &= \frac{ 1}{24}\left( 7474\zeta(9) -13122\zeta(3)\zeta(6) +6048\zeta(4)\zeta(5) +1953\zeta(2)\zeta(7) -544\zeta(3)^3\right)\\ {\cal I}(7,2) &= \frac{ 1}{72}\left( 276341\zeta(9) +88665\zeta(3)\zeta(6) +143163\zeta(4)\zeta(5) +59166\zeta(2)\zeta(7) +4032\zeta(3)^3\right) \end{align}\] The approximate numerical value of \({\cal I}(7,2)\) is \(9043.54574728044\); its integral estimate is \(8976.6033415307\).

We next present the first results we know of for order \(m+n = 10\). We originally obtained these results using the method described in Section 5. The eight basic sums \({\cal I}\) are obtained with two sums \({\cal I}(2,6)\) and \({\cal I}(2,8)\) assumed known: \[\begin{align} {\cal I}(1,9) &= \frac{1}{4}\left( 11\zeta(10) -4\zeta(3)\zeta(7) -2\zeta(5)^2\right) \\ {\cal I}(3,7) &= \frac{ 1}{160}\left( -1661\zeta(10) +1280\zeta(3)\zeta(7) +80\zeta(3)^2\zeta(4) -560\zeta(2)\zeta(3)\zeta(5) +720\zeta(5)^2 \right. \nonumber \\ &\left.+520 {\cal I}(2,8)\right) \end{align}\]

\[\begin{align} {\cal I}(4,6) &= \frac{ 1}{640}\left( -68823\zeta(10) +60000\zeta(3)\zeta(7) +1000\zeta(3)^2\zeta(4) -21680\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+23560\zeta(5)^2 +12120 {\cal I}(2,8) +1280\zeta(2) {\cal I}(2,6)\right) \\ {\cal I}(5,5) &= \frac{1}{256}\left( 64433\zeta(10) -57760\zeta(3)\zeta(7) +360\zeta(3)^2\zeta(4) +20560\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-22648\zeta(5)^2 -10920 {\cal I}(2,8) -1280\zeta(2) {\cal I}(2,6)\right) \\ {\cal I}(6,4) &= \frac{ 1}{128}\left( -271367\zeta(10) +176560\zeta(3)\zeta(7) -84648\zeta(3)^2\zeta(4) -400\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+121688\zeta(5)^2 +34376 {\cal I}(2,8) +15040\zeta(2) {\cal I}(2,6)\right) \\ {\cal I}(7,3) &= \frac{1}{2560}\left( 16614991\zeta(10) -10315520\zeta(3)\zeta(7) +5879160\zeta(3)^2\zeta(4) -705040\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-7710760\zeta(5)^2 -2021880 {\cal I}(2,8) -1008000\zeta(2) {\cal I}(2,6)\right) \\ {\cal I}(8,2) &= \frac{ 1}{480}\left( 18741581\zeta(10) +6689520\zeta(3)\zeta(7) -524640\zeta(3)^2\zeta(4) +1452480\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+4247040\zeta(5)^2 +485280 {\cal I}(2,8) +299520\zeta(2) {\cal I}(2,6)\right) \end{align}\]

We now present results for order \(m+n = 11\), which again are new in this study, and which again were originally found by us using the methods described below in Section 5. These formulas involve the two sums \({\cal I}(2,6)\) and \({\cal I}(3,8)\). \[\begin{align} {\cal I}(1,10) &= 6\zeta(11) -\zeta(2)\zeta(9) -\zeta(3)\zeta(8) -\zeta(4)\zeta(7) -\zeta(5)\zeta(6) \\ {\cal I}(2,9) &= \frac{1}{2}\left( 26\zeta(11) -2\zeta(2)\zeta(9) -9\zeta(3)\zeta(8) -5\zeta(4)\zeta(7) -7\zeta(5)\zeta(6) +2\zeta(3)^2\zeta(5)\right) \\ {\cal I}(4,7) &= \frac{ 1}{48}\left( -2877\zeta(11) -272\zeta(2)\zeta(9) +1190\zeta(3)\zeta(8) +1212\zeta(4)\zeta(7) +1018\zeta(5)\zeta(6) \right. \nonumber \\ &\left.+80\zeta(2)\zeta(3)^3 -576\zeta(3)^2\zeta(5) +176 {\cal I}(3,8)\right) \\ {\cal I}(5,6) &= \frac{ 1}{576}\left( -781671\zeta(11) -88016\zeta(2)\zeta(9) +296660\zeta(3)\zeta(8) +411984\zeta(4)\zeta(7) \right. \nonumber \\ &\left.+220080\zeta(5)\zeta(6) +21120\zeta(2)\zeta(3)^3 -141120\zeta(3)^2\zeta(5) +8640\zeta(3) {\cal I}(2,6) \right. \nonumber \\ &\left.+27840 {\cal I}(3,8)\right) \\ {\cal I}(6,5) &= \frac{1}{192}\left( 734643\zeta(11) 83472\zeta(2)\zeta(9) -271244\zeta(3)\zeta(8) -395088\zeta(4)\zeta(7) \right. \nonumber \\ &\left.-205424\zeta(5)\zeta(6) -19360\zeta(2)\zeta(3)^3 +130176\zeta(3)^2\zeta(5) -9120\zeta(3) {\cal I}(2,6) \right. \nonumber \\ &\left.-25600 {\cal I}(3,8)\right) \\ {\cal I}(7,4) &= \frac{1}{1152}\left( 16370805\zeta(11) 1684144\zeta(2)\zeta(9) +5889744\zeta(3)\zeta(8) -10724760\zeta(4)\zeta(7) \right. \nonumber \\ &\left.-10480104\zeta(5)\zeta(6) +844032\zeta(2)\zeta(3)^3 -2330496\zeta(3)^2\zeta(5) -1431360\zeta(3) {\cal I}(2,6) \right. \nonumber \\ &\left.-630336 {\cal I}(3,8)\right) \\ {\cal I}(8,3) &= \frac{1}{72}\left( 2824380\zeta(11) 277304\zeta(2)\zeta(9) +1926401\zeta(3)\zeta(8) -1998972\zeta(4)\zeta(7) \right. \nonumber \\ &\left.-2270310\zeta(5)\zeta(6) +243648\zeta(2)\zeta(3)^3 -803808\zeta(3)^2\zeta(5) -341280\zeta(3) {\cal I}(2,6) \right. \nonumber \\ &\left.-113760 {\cal I}(3,8)\right) \\ {\cal I}(9,2) &= \frac{ 1}{64}\left( 7739347\zeta(11) +2048432\zeta(2)\zeta(9) +5357920\zeta(3)\zeta(8) \right. \nonumber \\ &\left.+8811792\zeta(4)\zeta(7) +10526056\zeta(5)\zeta(6) -294208\zeta(2)\zeta(3)^3 +2064192\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.+540096\zeta(3) {\cal I}(2,6) +199936 {\cal I}(3,8)\right) \end{align}\]

Finally, we present results for order \(m+n = 12\), which as before are new to this study, having been originally obtained by us using the methods described in Section 5. These results involve the two sums \({\cal I}(2,10)\) and \({\cal I}(4,8)\): \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{11}} &= \frac{1}{4}\left( 13\zeta(12) -4\zeta(3)\zeta(9) -4\zeta(5)\zeta(7)\right) \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{9}} &= \frac{1}{22112}\left( 355355\zeta(12) -221120\zeta(3)\zeta(9) -265344\zeta(5)\zeta(7) \right. \nonumber \\ &\left.-33168\zeta(3)^2\zeta(6) +5528\zeta(3)^4 +49752\zeta(2)\zeta(5)^2 +99504\zeta(2)\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-82920 {\cal I}(2,10)\right) \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{7}} &= \frac{ 1}{265344}\left( 3612841\zeta(12) -884480\zeta(3)\zeta(9) -597024\zeta(5)\zeta(7) \right. \nonumber \\ &\left.+364848\zeta(3)^2\zeta(6) +221120\zeta(3)^4 +364848\zeta(2)\zeta(5)^2 +729696\zeta(2)\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-3250464\zeta(3)\zeta(4)\zeta(5) -1028208 {\cal I}(2,10) +663360 {\cal I}(4,8)\right) \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{6}} &= \frac{ 1}{530688}\left( -4262917573\zeta(12) +2820739392\zeta(3)\zeta(9) +2446737024\zeta(5)\zeta(7) \right. \nonumber \\ &\left.+112663404\zeta(3)^2\zeta(6) -41128320\zeta(3)^4 -402626352\zeta(2)\zeta(5)^2 \right. \nonumber \\ &\left.-741769152\zeta(2)\zeta(3)\zeta(7) -205077744\zeta(3)\zeta(4)\zeta(5) +52538112\zeta(4) {\cal I}(2,6) \right. \nonumber \\ &\left.+84213552\zeta(2) {\cal I}(2,8) +519676224 {\cal I}(2,10) -22554240 {\cal I}(4,8)\right) \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k^{5}} &= \frac{ 1}{1061376}\left( -29991036967\zeta(12) +19798731008\zeta(3)\zeta(9) +17219233536\zeta(5)\zeta(7) \right. \nonumber \\ &\left.+722473668\zeta(3)^2\zeta(6) -292232192\zeta(3)^4 -2832315024\zeta(2)\zeta(5)^2 \right. \nonumber \\ &\left.-5220245184\zeta(2)\zeta(3)\zeta(7) -1329671952\zeta(3)\zeta(4)\zeta(5) +381697344\zeta(4) {\cal I}(2,6) \right. \nonumber \\ &\left.+589494864\zeta(2) {\cal I}(2,8) +3662808576 {\cal I}(2,10) -167166720 {\cal I}(4,8)\right) \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{k^{4}} &= \frac{ 1}{199008}\left( -6469168763\zeta(12) -4417645920\zeta(3)\zeta(9) +2316436536\zeta(5)\zeta(7) \right. \nonumber \\ &\left.-7185432600\zeta(3)^2\zeta(6) +210815808\zeta(3)^4 +2292190728\zeta(2)\zeta(5)^2 \right. \nonumber \\ &\left.+3705761136\zeta(2)\zeta(3)\zeta(7) +4396086720\zeta(3)\zeta(4)\zeta(5) +2171077776\zeta(4) {\cal I}(2,6) \right. \nonumber \\ &\left.+241230864\zeta(2) {\cal I}(2,8) -842782296 {\cal I}(2,10) +116552352 {\cal I}(4,8)\right) \\ \sum_{k=1}^\infty \frac{H(k)^{9}}{k^{3}} &= \frac{ 1}{176896}\left( 4340755723\zeta(12) -37498812096\zeta(3)\zeta(9) -8003239392\zeta(5)\zeta(7) \right. \nonumber \\ &\left.-29417337684\zeta(3)^2\zeta(6) +1136645248\zeta(3)^4 +12010630320\zeta(2)\zeta(5)^2 \right. \nonumber \\ &\left.+20062394496\zeta(2)\zeta(3)\zeta(7) +18880585488\zeta(3)\zeta(4)\zeta(5) +8292663360\zeta(4) {\cal I}(2,6) \right. \nonumber \\ &\left.+375428592\zeta(2) {\cal I}(2,8) -7045878240 {\cal I}(2,10) +635233536 {\cal I}(4,8)\right) \\ \sum_{k=1}^\infty \frac{H(k)^{10}}{k^{2}} &= \frac{ 1}{176896}\left( 702828643635\zeta(12) +39514453568\zeta(3)\zeta(9) +93510608736\zeta(5)\zeta(7) \right. \nonumber \\ &\left.-23538514220\zeta(3)^2\zeta(6) +2706951040\zeta(3)^4 +35094519056\zeta(2)\zeta(5)^2 \right. \nonumber \\ &\left.+62104868800\zeta(2)\zeta(3)\zeta(7) +96955381936\zeta(3)\zeta(4)\zeta(5) +16400028160\zeta(4) {\cal I}(2,6) \right. \nonumber \\ &\left.+954077520\zeta(2) {\cal I}(2,8) -12973442080 {\cal I}(2,10) +1115329280 {\cal I}(4,8)\right) \end{align}\]

2.1 Asymptotics of the sum \({\cal I} (m,2)\)↩︎

The summands of the Euler sums \({\cal I} (m,n)\) are always positive, and increase as \(m\) increases, while decreasing as \(n\) increases. The values of sums depending on the \({\cal I} (m,n)\) discussed in this paper tend to be dominated by the lowest sum \({\cal I} (m,2)\) for large values of \(m\), and so it is valuable to have asymptotic approximations for it. The value of the sum can be well estimated by an integral, given that the maximum of the truncated summand \((\log k+\gamma)^m/k^2\) occurs for \(k=\exp (m/2)\), large enough for the discrete sum to be well approximated by the corresponding integral. For a general positive integer \(q\), the result follows from the recursion \[\begin{align} {\cal N}_{q+1} &= \int_1^\infty \frac{(\log k+\gamma)^{q+1} \, {\rm d}k}{k^2}=\gamma^{q+1}+(q+1) {\cal N}_q,{\cal N}_1=1+\gamma , \label{intrecur} \end{align}\tag{28}\] where \(\gamma = 0.5772156649\ldots\) is Euler’s constant. This recurrence can be solved exactly, giving \[\begin{align} \int_1^\infty \frac{(\log k+\gamma)^m}{k^2} {\rm d}k &= m! \left[e^\gamma\right]_m . \label{form:pow8c} \end{align}\tag{29}\] Here we have introduced the notation for the truncated exponential: \[\left[e^\gamma\right]_m =1+\sum_{q=1}^m \frac{\gamma^q}{q!} . \label{pow8cc}\tag{30}\] Note that for large \(m\), \({\cal N}_m/m! \rightarrow \exp(\gamma)\).

Although the integral in equation (29 ) is exactly evaluated, its use in approximating the sums \({\cal I} (m,2)\) for \(m\) large depends on two approximations: the sum is well approximated by an integral, and the two-term asymptotic series of the harmonic number function gives a sufficiently accurate representation for the integrand. These approximations are tested in Table ¿tbl:tab:tabIm2?, which shows that the integral approximation gains relative accuracy rapidly as \(m\) increases, until at \(m=9\) it is accurate to two parts in 1000.

3 Mixed Euler sums↩︎

In the previous sections, we have focused on the \({\cal I}\) sums, whose denominators are powers of \(k\), and on the \(s_h\) sums, which have powers of \(k+1\). But one is immediately led to consider more general denominators, which have not been previously studied in the literature in any detail. To that end we now consider “mixed Euler sums,” namely sums such as \[\begin{align} M(m,n_0,n_1,n_2, \ldots, n_t) &= \sum_{k=1}^\infty \frac{H(k)^m}{k^{n_0} (k+1)^{n_1} (k+2)^{n_2} \cdots (k+t)^{n_t}}, \label{form:Mixed} \end{align}\tag{31}\] for nonnegative integers \(m\) and \((n_i)\), with \(m \geq 1\) and \(n_0 + n_1 + \cdots + n_t \geq 2\), where \(H(k) = H_k = \sum_{j=1}^k 1/j\) is the harmonic function as before, and where \(r = m + n_0 + n_1 + \cdots + n_t\) is the order. It is clear that the \(s_h\) and \({\cal I}\) sums are merely special cases: \(s_h(m,n) = M(m,0,n)\) and \({\cal I}(m,n) = M(m,n)\), so hereafter we will use the \(M\) notation. We first demonstrate, by means of examples, why Euler sums with more complicated denominators can be reduced to the basic \(M(m,n)\) cases.

Note: We conjecture that the representation of a order-12 or less mixed Euler sum as a rational linear combination of the constants in the list in Theorem 1 above is unique, since integer relation computations on this set rule out any relations with reasonable-sized coefficients (see next paragraph for details), but we have no proof of this. We also conjecture that a result similar to Theorem 1 applies for all higher orders: most likely it only remains to identify the appropriate “atoms,” akin to the list in Theorem 1.

We should also clarify that Theorem 1 relies in part on some results in the previous section that were obtained using the computational techniques described below in Section 5.

Note that the above list includes the constants \(M(2,6), \, M(2,8), \, M(3,8), \, M(2,10)\) and \(M(4,8)\). These constants appear to be linearly independent from the rest of the set, as indicated by the fact that a multipair PSLQ computer run (see Section 5) with the full set of order 8 constants shown above finds no integer relation with Euclidean norm less than \(5.88 \cdot 10^{22}\); the full set of order 10 constants above produces no integer relation with Euclidean norm less than \(1.28 \cdot 10^{13}\); and the full set of order 12 constants produces no integer relation with Euclidean norm less than \(2.13 \cdot 10^6\). Nevertheless, the question of whether \(M(2,6), \, M(2,8), \, M(3,8), \, M(2,10)\) and \(M(4,8)\), singly or collectively, can be expressed analytically in terms of zetas or other well-known mathematical constants remains open. As an aid to further research, we include 400-digit values of these constants in Appendix 1 (Section 7).

Sketch of proof: We first observe (see Section 2 above) that each of the basic Euler sums \(M(m,n) = {\cal I}(m,n)\) with order \(m+n \leq 12\) is reducible to a rational linear sum of the above-listed “atomic” constants. We now argue that any general mixed Euler sum 31 of order 12 or less can be reduced to a rational linear combination of the basic Euler sums \(M(m,n)\) of the same order or less, and thus to a rational linear combination of the constants in Theorem 1, by the application (possibly repeated) of these two algebraic techniques:

1. Changing sums with expressions involving \((k+1), \, (k+2)\) or \((k+w)\) for any integer \(w > 0\) to sums involving only \(k\), by means of a process akin to “completing the square” of elementary algebra.

2. Applying a partial fraction decomposition: Recall that any rational function can be written uniquely as the sum of terms based on the factorization of the denominator polynomial, as in the example \[\begin{align} \frac{1}{(k + 1)(k + 2)^2} &= \frac{1}{k+1} - \frac{1}{k+2} - \frac{1}{(k+2)^2}. \end{align}\] This can be produced in Wolfram Mathematica by the command: Apart[1/((k+1)*(k+2)^2)]``.

To illustrate these techniques, note that one can write \(M(2,0,2) = \sum_{k=1}^\infty H(k)^2 / (k+1)^2\) as \[\begin{align} M(2,0,2) &= \sum_{k=1}^\infty \frac{H(k)^2}{(k+1)^2} \nonumber \\ &= \frac{1}{2^2} + \frac{(1+1/2)^2}{3^2} + \frac{(1+1/2+1/3)^2}{4^2} + \frac{(1+1/2+1/3+1/4)^2}{5^2} + \cdots \nonumber \\ &= \left(\frac{(1+1/2)^2}{2^2} - \frac{2/2}{2^2} - \frac{1/4}{2^2}\right) + \left(\frac{(1+1/2+1/3)^2}{3^2} - \frac{2/3(1+1/2)}{3^2} - \frac{1/9}{3^2}\right) \nonumber \\ & + \left(\frac{(1+1/2+1/3+1/4)^2}{4^2} - \frac{2/4(1+1/2+1/3)}{4^2} - \frac{1/16}{4^2}\right) + \cdots \nonumber \\ &= \left(\frac{(1+1/2)^2}{2^2} + \frac{(1+1/2+1/3)^2}{3^2} + \cdots\right) - 2 \left(\frac{1}{2^3} + \frac{(1+1/2)}{3^3} \cdots\right) - \left(\frac{1}{2^4} + \frac{1}{3^4} + \cdots\right) \nonumber \\ &= \left(\sum_{k=1}^\infty \frac{H(k)^2}{k^2} - 1\right) - 2 \sum_{k=1}^\infty \frac{H(k)}{(k+1)^3} - (\zeta(4) - 1) \nonumber \\ &= M(2,2) - 2 \, M(1,0,3) - \zeta(4). \label{form:M202} \end{align}\tag{32}\] Note, crucially, that this manipulation rewrites the mixed Euler sum \(M(2,0,2)\) (of order 4) to an expression involving \(M(2,2)\) (of order 4), the mixed sum \(M(1,0,3) = \sum_{k=1}^\infty H(k)/(k+1)^3\), (also of order 4), and the constant \(\zeta(4)\) (again of order 4). A similar manipulation can now be performed on \(M(1,0,3)\): \[\begin{align} M(1,0,3) &= \sum_{k=1}^\infty \frac{H(k)}{(k+1)^3} \; = \; \frac{1}{2^3} + \frac{(1+1/2)}{3^3} + \frac{(1+1/2+1/3)}{4^3} + \cdots \nonumber \\ &= \left(\frac{(1+1/2)}{2^3} - \frac{1/2}{2^3}\right) + \left(\frac{(1+1/2+1/3)}{3^3} - \frac{1/3}{3^3}\right) + \cdots \nonumber \\ &= \left(\sum_{k=1}^\infty \frac{H(k)}{k^3} - 1\right) - (\zeta(4) - 1) \nonumber \\ &= M(1,3) - \zeta(4) \; = \; 5/4 \, \zeta(4) - \zeta(4) \; = \; 1/4 \, \zeta(4), \label{form:M103} \end{align}\tag{33}\] so that \(M(2,0,2) = M(2,2) - 2 \, M(1,0,3) - \zeta(4) = 17/4 \, \zeta(4) - 1/2 \, \zeta(4) - \zeta(4) = 11/4 \, \zeta(4)\), which is of order 4. Note that none of these algebraic manipulations increased the order.

A second example of this technique is \(M(2,1,1) = \sum_{k \geq 1} H(k)^2/(k(k+1))\). Note that by employing a manipulation similar to that used above in 32 and 33 , combined with the partial fraction decomposition \[\begin{align} \frac{1}{k(k+1)} &= \frac{1}{k} - \frac{1}{k+1}, \end{align}\] this can be written \[\begin{align} M(2,1,1) &= \sum_{k=1}^\infty \frac{H(k)^2}{k(k+1)} \; = \; \sum_{k=1}^\infty \left(\frac{H(k)^2}{k} - \frac{H(k)^2}{k+1}\right) \nonumber \\ &= \left(\frac{1}{1} + \frac{(1+1/2)^2}{2} + \frac{(1+1/2+1/3)^2}{3} + \cdots\right) - \left(\frac{1}{2} + \frac{(1+1/2)^2}{3} + \frac{(1+1/2+1/3)^2}{4} + \cdots\right) \nonumber \\ &= \left(\frac{1}{1} + \frac{(1+1/2)^2}{2} + \frac{(1+1/2+1/3)^2}{3} + \cdots\right) \nonumber \\ &- \left[\left(\frac{(1+1/2)^2}{2} - \frac{2/2}{2} - \frac{1/4}{2}\right) + \left(\frac{(1+1/2+1/3)^2}{3} - \frac{2/3 \,(1+1/2)}{3} - \frac{1/9}{3}\right) \right. \nonumber \\ &\left. + \left(\frac{(1+1/2+1/3+1/4)^2}{4} - \frac{2/4 \, (1+1/2+1/3)}{4} - \frac{1/16}{4}\right) + \cdots\right] \nonumber \\ &= 1 + 2 \sum_{k=1}^\infty \frac{H(k)}{(k+1)^2} + (\zeta(3) - 1) \; = \; 2 M(1,2) + \zeta(3) \; = \; 3 \, \zeta(3). \label{form:M111} \end{align}\tag{34}\] Note again that none of these operations increased the order; in fact, in this case the order of the final result, namely 3, is less than the order of the original problem, namely 4.

Consider now a more complicated sum such as \(M(2,2,2) = \sum_{k \geq 1} H(k)^2 / (k^2 (k+1)^2\). Sums like this can be readily reduced by means of a partial fraction decomposition, which in this case is: \[\begin{align} \frac{1}{k^2(k+1)^2} &= -2\left(\frac{1}{k} - \frac{1}{k+1}\right) + \frac{1}{k^2} + \frac{1}{(k+1)^2}, \end{align}\] so that \[\begin{align} M(2,2,2) &= \sum_{k=1}^\infty \frac{H(k)^2}{k^2 (k+1)^2} \nonumber \\ &= -2 \sum_{k=1}^\infty H(k)^2 \left(\frac{1}{k} - \frac{1}{k+1}\right) + \sum_{k=1}^\infty \frac{H(k)^2}{k^2} + \sum_{k=1}^\infty \frac{H(k)^2}{(k+1)^2} \nonumber \\ &= -6 \zeta(3) + 17/4 \, \zeta(4) + 11/4 \, \zeta(4) \nonumber \\ &= 7 \zeta(4) - 6 \zeta(3), \end{align}\] where we have employed results from 32 , 33 and 34 above.

One example involving \((k+2)\) is \(M(2,0,0,2) = \sum_{k \geq 1} H(k)^2/(k+2)^2\). This can be reduced as follows (omitting details of some intermediate evaluations using the above techniques): \[\begin{align} M(2,0,0,2) &= \sum_{k=1}^\infty \frac{H(k)^2}{(k+2)^2} \nonumber \\ &= \frac{1^2}{3^2} + \frac{(1+1/2)^2}{4^2} + \frac{(1+1/2+1/3)^2}{5^2} + \frac{(1+1/2+1/3+1/4)^2}{6^2} + \cdots \nonumber \\ &= \left(\frac{(1+1/2+1/3)^2}{3^2} - \frac{2(1/2+1/3)}{3^2} - \frac{(1/2+1/3)^2}{3^2} \right) \nonumber \\ &+ \left(\frac{(1/2+1/3+1/4)^2}{4^2} - \frac{2(1+1/2)(1/3+1/4)}{4^2} - \frac{(1/3+1/4)^2}{4^2} \right) \nonumber \\ &+ \left(\frac{1/2+1/3+1/4+1/5)^2}{5^2} - \frac{2(1+1/2+1/3)(1/4+1/5)}{5^2} - \frac{(1/4+1/5)^2}{5^2}\right) + \cdots \nonumber \end{align}\]

\[\begin{align} &= \sum_{k=3}^\infty \frac{H(k)^2}{k^2} - 2 \sum_{k=1}^\infty \frac{H(k)}{(k+1)(k+2)^2} - 2 \sum_{k=1}^\infty \frac{H(k)}{(k+2)^3} - \sum_{k=1}^\infty \frac{1}{(k+1)^2 (k+2)^2} \nonumber \\ &- 2 \sum_{k=1}^ \infty \frac{1}{(k+1)(k+2)^3} - \sum_{k=1}^\infty \frac{1}{(k+2)^4} \nonumber \\ &= \left(-\frac{25}{16} + \frac{17}{4} \, \zeta(4)\right) - 2\left(3 - \zeta(2) - \zeta(3)\right) - 2 \left(-3 + \zeta(2) + \zeta(3) + \frac{1}{4} \zeta(4) \right) \nonumber \\ &- \left(-\frac{13}{4} + 2 \zeta(2)\right) - 2 \left(\frac{23}{8} - \zeta(2) - \zeta(3)\right) - \left(-\frac{17}{16} + \zeta(4)\right) \nonumber \\ &= -3 + 2 \zeta(3) + \frac{11}{4} \zeta(4) \label{M2002} \end{align}\tag{35}\]

The same techniques work for denominators involving \((k+w)\) for any integer \(w > 2\). For example, after rather laborious effort one can deduce that \[\begin{align} M(2,0,0,0,0,2) &= \sum_{k=1}^\infty \frac{H(k)^2}{(k+4)^2} \; = \; \frac{1}{3456}\left(-1045 + 288 \, \zeta(2) + 648 \, \zeta(3) \right). \end{align}\]

We should note, however, that the algebraic manipulations required in these evaluations grow very sharply in complexity with increasing powers of \(H(k)\) in the numerator and increasing terms in the denominator. Thus we have found, quite frankly, that in most cases these analytic formulas are more easily obtained by the computational methods we describe below in Section 5.

4 Euler sum analogues to the Stieltjes constants↩︎

As an application of these techniques, we address a finite Euler sum result from Choi and Srivastava[4]: \[\sum_{k=1}^{n-1} \frac{H_k}{k}=\frac{1}{2} H_{n-1}^2+\frac{1}{2} H_{n-1}^{(2)}. \label{ChoiSri}\tag{36}\] (Note that in this section it is more convenient to use the subscript notation for the harmonic function: \(H_k = H(k)\).) We note also the following result from [9]: \[\sum_{k=1}^n\frac{H_k^{(2)}}{k}=H_n^{(2)} H_n-\sum_{k=1}^n \frac{H_k}{k^2}+H_n^{(3)}. \label{harm9}\tag{37}\] This is easily generalised to an arbitrary harmonic number of order \(p\): \[\sum_{k=1}^n\frac{H_k^{(p)}}{k}=H_n^{(p)} H_n-\sum_{k=1}^n \frac{H_k}{k^p}+H_n^{(p+1)}. \label{harm10}\tag{38}\]

We continue with the sums over a finite range, commencing with a result given in [10]: \[\sum_{k=1}^n \frac{H_k^2}{k}=\frac{1}{3} H_n^3-\frac{1}{3} H_n^{(3)}+\sum_{k=1}^n \frac{H_k}{k^2}. \label{form:1onk1}\tag{39}\] The method used to derive 39 employs Abel’s summation formula and may easily be generalised to higher values of \(p\). When this was done, a pattern emerged for all the sums: \[\sum_{k=1}^n \frac{H_k^{p-1}}{k}=\frac{1}{p} H_n^p+ {\cal D}_{p,1} H_n^{(p)}+\sum_{q=2}^{p-1} {\cal D}_{p,q} \sum_{k=1}^n \frac{H_k^{q-1}}{k^{p-q+1}}. \label{form:1onk2}\tag{40}\] Employing this pattern, it is easy to evaluate the coefficients \({\cal D}_{p,q}\) by choosing the same number of values of \(n\) as the number of unknowns, and solving linear equations for the \(p-1\) unknowns. The values obtained can easily be checked for other values of \(n\). Note that the coefficients in the linear equations are exactly known, and the values for the \({\cal D}_{p,q}\) are also exact. Some values are given in Table 1.

The lists of coefficients in Table 1 have some evident properties. The sum of the \({\cal D}_{p,q}\) over \(q\) when combined with \(1/p\) from the first term on the right-hand side in equation (40 ) is required to be unity, so that the results for \(n=1\) on both sides of equation (40 ) match. For \(p\) even, \({\cal D}_{p,1}=1/p\) and later coefficients show an even symmetry. For \(p\) odd, later coefficients show an odd symmetry. For \(p\) odd, the second coefficient in Table 1 is 1, while for \(p\) even it is -1. There are \(p-1\) coefficients \({\cal D}\), with the first two being \(1/p,\pm 1\). The rest of the \({\cal D}\)’s fall into \((p-3)/2\) pairs which combine subtractively for \(p\) odd, or \((p-4)/2\) additive pairs and a central element for \(p\) even.

We take the limit as \(n\rightarrow \infty\) in equation (40 ) to define a set of harmonic sum Stieltjes constants \(\gamma_p^H\), where: \[\lim_{n\rightarrow \infty}\left[\sum_{k=1}^n \frac{H_k^{p-1}}{k}-\frac{1}{p} H_n^p\right]=\gamma_p^H\label{form:1onk3},\tag{41}\] and \[\begin{align} \gamma_p^H &=& {\cal D}_{p,1} \zeta (p)+\sum_{q=2}^{p-1} {\cal D}_{p,q} \sum_{k=1}^\infty \frac{H_k^{q-1}}{k^{p-q+1}}\nonumber \\ &=& {\cal D}_{p,1} \zeta (p)+\sum_{q=2}^{p-1} {\cal D}_{p,q} M_{q-1,p-q+1}. \label{form:1onk4} \end{align}\tag{42}\] The most slowly convergent by direct summation of the terms in 42 occurs for \(q=p-1\), where the summand goes to zero as \(\log(k)^{p-2}/k^2\).

By applying formula 42 , together with the computational techniques described below in Section 5, we were able to obtain these results: \[\begin{align} \gamma_2^H &= \frac{1}{2}\left( \zeta(2)\right) \\ \gamma_3^H &= \frac{ 1}{3}\left( 5\zeta(3)\right) \\ \gamma_4^H &= \frac{ 1}{8}\left( 43\zeta(4)\right) \\ \gamma_5^H &= \frac{ 1}{5}\left( 79\zeta(5) +15\zeta(2)\zeta(3)\right) \\ \gamma_6^H &= \frac{ 1}{24}\left( 2187\zeta(6) +272\zeta(3)^2\right) \\ \gamma_7^H &= \frac{1}{56}\left( 18311\zeta(7) +4060\zeta(2)\zeta(5) +8358\zeta(3)\zeta(4)\right) \\ \gamma_8^H &= \frac{1}{576}\left( 1926401\zeta(8) +48384\zeta(2)\zeta(3)^2 +440064\zeta(3)\zeta(5)\right) \\ \gamma_9^H &= \frac{1}{36}\left( 501978\zeta(9) +266355\zeta(3)\zeta(6) +241794\zeta(4)\zeta(5) +105273\zeta(2)\zeta(7) +12104\zeta(3)^3\right) \\ \gamma_{10}^H &= \frac{1}{80}\left( 17061619\zeta(10) +3161210\zeta(3)\zeta(7) +705180\zeta(3)^2\zeta(4) +928080\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+1770112\zeta(5)^2 +37320\zeta(2) M(2,6)\right) \\ \gamma_{11}^H &= \frac{1}{264}\left( 230253219\zeta(11) +49094276\zeta(2)\zeta(9) +165822855\zeta(3)\zeta(8) +130449891\zeta(4)\zeta(7) \right. \nonumber \\ &\left.+156493260\zeta(5)\zeta(6) +805200\zeta(2)\zeta(3)^3 +19281504\zeta(3)^2\zeta(5) +1849320\zeta(3) M(2,6) \right. \nonumber \\ &\left.+1232880 M(3,8)\right) \end{align}\]

Using the integral estimate \({\cal N}_n\rightarrow [ \exp(\gamma)]_n n!\) and replacing \(n\) by \(p-2\), we multiply this by \((p-1)/2\), the final \({\cal D}_{p,q}\) in each line of Table 2. This gives the estimate for \(\gamma_p^H\): \[\gamma_p^H\approx [\exp(\gamma)]_{p-2} \frac{(p-1)! }{2} . \label{form:gammapest}\tag{43}\]

The approximation 43 is compared with the \(\gamma_p^H\) for \(p\) ranging from 3 to 11 in Table ¿tbl:tabgapH?. The trend is clearly for the relative accuracy to improve as \(p\) increases.

We turn now to equivalent expressions for the case when the denominator in the basic sum is \(k+1\) rather than \(k\). From [4], \[\sum_{k=1}^{n-1} \frac{H_k}{k+1}=\frac{1}{2} H_{n}^2-\frac{1}{2} H_{n}^{(2)}. \label{fstkp1}\tag{44}\] The equivalent of equation (39 ) is \[\sum_{k=1}^{n-1} \frac{H_k^2}{k+1}=\frac{1}{3} H_n^3-\frac{1}{3} H_n^{(3)}-\sum_{k=1}^n \frac{H_k}{(k+1)^2}. \label{1onk1p}\tag{45}\] Once again, this may be extended to higher powers of \(H_k\) in the numerator, giving results of the following form: \[\sum_{k=1}^{n-1} \frac{H_k^{p-1}}{k+1}=\frac{1}{p} H_n^p + {\cal E}_{p,1} H_n^{(p)}+\sum_{q=2}^{p-1} {\cal E}_{p,q} \sum_{k=1}^n \frac{H_k^{q-1}}{(k+1)^{p-q+1}}. \label{1onk2p}\tag{46}\] The sums on the right-hand side are the quantities \(s_h\) studied inter alia in [6], [7], and connected with the \({\cal I}\) by (10 ). The coefficients \({\cal E}_{p,q}\) are given for \(p\) up to 21 in Table 2. Note that all the \({\cal E}_{p,q}\) are negative, unlike the alternating sign behaviour of the \({\cal D}_{p,q}\).

We can define an alternate set of Stieltjes-like constants from equation (46 ): \[\lim_{n\rightarrow \infty} \left[\sum_{k=1}^{n-1} \frac{H_k^{p-1}}{k+1}-\frac{1}{p} H_n^p \right]=\gamma^h_p, \label{hgas}\tag{47}\] where \[\gamma^h_p= {\cal E}_{p,1} \zeta (p)+\sum_{q=2}^{p-1} {\cal E}_{p,q} s_h (q-1,p-q+1). \label{hgasa}\tag{48}\] The \(\gamma^h_p\) would then all be negative, and by virtue of (8 ) with a modulus well approximated asymptotically by (43 ).

5 Computational techniques↩︎

We initially obtained many of the formulas presented above and in Appendix 2 (Section 8) by a computational procedure that utilizes advanced techniques to produce a very high-precision numerical value of the sum, then employs an integer relation algorithm to identify the numerical value as a rational linear sum of constants from the list in Theorem 1. We present here a brief summary of these techniques, which are based in part on schemes described in [6], [11].

In this way, we have found formulas for the much more numerous set of mixed Euler cases \[\begin{align} M(m,n,p,q) &= \sum_{k=1}^\infty \frac{H_k^m}{k^n (k+1)^p (k+2)^q} \label{form:Mnpq} \end{align}\tag{49}\] for orders \(r = m + n + p + q = 3\) through \(11\), and also a selection of results of order 12. As noted above, this class includes the cases \(s_h(m,n)\) and \({\cal I}(m,n)\) as subsets. We present the full collection of these formulas in Appendix 2 (Section 8).

5.1 Computing Euler sums to high precision↩︎

One key tool for these computations is the Euler-Maclaurin summation formula [12], which approximates a summation as an integral with high-order corrections (here \(f(t)\) is assumed to have \((2s+2)\)-th order derivatives on \([a,b]\)): \[\begin{align} \sum_{j=a}^b f(j) &= \int_a^b f(t) \, {\rm d}t \, + \, \frac{1}{2} \left(f(a) + f(b)\right) \, + \,\sum_{j=1}^s \frac{B_{2j} \left(D^{2j-1} f(b) - D^{2j-1} f(a)\right)}{(2j)!} \, + \, R_s(a,b), \label{form:eulermac} \end{align}\tag{50}\] where \(B_k\) is the \(k\)-th Bernoulli number [13], \(D^k f(a)\) is the \(k\)-th derivative of \(f(t)\) evaluated at \(t = a\), and \[\begin{align} R_s(a,b) &= \frac{-1}{(2s+2)!} \int_a^b B_{2s+2} (t - [t]) D^{2s+2} f(t) \, {\rm d}t, \end{align}\] where \([\cdot]\) denotes greatest integer and \(B_{k}(\cdot)\) is the \(k\)-th Bernoulli polynomial [13] (note \(B_k = B_k(0)\)).

Applying the Euler-Maclaurin summation formula to the harmonic function \(H(t) = \sum_{j=1}^t 1/j\) yields \[\begin{align} H(t) &= \gamma + \log(t) + \frac{1}{2t} + \sum_{j=1}^s \frac{B_{2j}}{2j t^{2j}} + R_s(t), \end{align}\] where \(\gamma = 0.5772156649\ldots\) is Euler’s constant and \(|R_s(t)| \leq |B_{2s+2}|/((2s+2) t^{2s+2})\); see [6] for full details. In the computations for the present study, we set \(s = 21\), so that \(H(t)\) is approximated by \[\begin{align} \hat{H}(t) &= \gamma + \log(t) + \frac{1}{2 t} - \frac{1}{12 t^2} + \frac{1}{120 t^4} - \frac{1}{252 t^6} + \frac{1}{240 t^8} - \frac{1}{132 t^{10}} + \frac{691}{32760 t^{12}} - \frac{1}{12 t^{14}} \nonumber \\ & + \frac{3617}{8160 t^{16}} - \frac{43867}{14364 t^{18}} + \frac{174611}{6600 t^{20}} - \frac{77683}{276 t^{22}} + \frac{236364091}{65520 t^{24}} - \frac{657931}{12 t^{26}} + \frac{3392780147}{3480 t^{28}} \nonumber \\ & - \frac{1723168255201}{85932 t^{30}} + \frac{7709321041217}{16320 t^{32}} - \frac{151628697551}{12 t^{34}} + \frac{26315271553053477373}{69090840 t^{36}} \nonumber \\ & - \frac{154210205991661}{12 t^{38}} + \frac{261082718496449122051}{541200 t^{40}} - \frac{1520097643918070802691}{75852 t^{42}}, \label{form:happrox} \end{align}\tag{51}\] which approximates \(H(t)\) to within roughly \(t^{-44}\) for large \(t\). The expression 51 can be obtained using Wolfram Mathematica with the command Series[HarmonicNumber[t],{t,Infinity,42}].

Given \(M(m,n,p,q)\), denote \(\hat{G}(t) = \hat{H}(t)^m /(t^n (t+1)^p (t+2)^q)\). Using the Euler-Maclaurin summation formula 50 once again, one can write \[\begin{align} M(m,n,p,q) &= \sum_{j=1}^k \frac{H(j)^m}{j^n (j+1)^p (j+2)^q} + \sum_{j=k+1}^\infty \frac{H(j)^m}{j^n (j+1)^p (j+2)^q} \; \approx \; \sum_{j=1}^k \frac{H(j)^m}{j^n (j+1)^p (j+2)^q} + \sum_{j=k+1}^\infty \hat{G}(j) \nonumber \\ & \approx \sum_{j=1}^k \frac{H(j)^m}{j^n (j+1)^p (j+2)^q} \, + \int_{k+1}^\infty \hat{G}(t) \, {\rm d}t \, + \frac{1}{2} \hat{G}(k + 1) \, - \sum_{j=1}^{s} \frac{B_{2j} D^{2j-1} \hat{G} (k+1)}{(2j)!}, \label{form:Mapprox} \end{align}\tag{52}\] where \(s = 21\), which is accurate to within roughly \(k^{-44}\). Initially we set \(k = 10^8 = 100,000,000\), so the approximation in the second line of 52 is correct to within roughly \(10^{-354}\), which was sufficient for our early investigations. For larger cases, and for all runs listed in Appendix 2, we set \(k = 10^9 = 1,000,000,000\), so this approximation is correct to within roughly \(10^{-396}\).

We evaluated the first term of 52 (the explicit summation) using an arbitrary precision package [14]. Using \(k = 10^8\) and a working precision of 360 digits (producing roughly 350 good digits) required 5–9 minutes CPU time per case on a 2024 Apple Mac Studio system with an M4 processor; using \(k = 10^9\) and a working precision of 420 digits (producing roughly 400 good digits) required 50–90 minutes per case. For our 400-digit computations, we evaluated the second term (the integral) using the exp-sinh quadrature algorithm [14], [15], with the arbitrary precision software set to 400 digits; this required only 3–4 seconds per case (we first tried to evaluate these integrals using Wolfram Mathematica version 14.2, but this failed for larger \(m\)). The third term is straightforward. The fourth term, which involves the symbolic expansion and numerical evaluation to 400-digit accuracy of high-order derivatives of the approximation function \(\hat{G}(t) = \hat{H}(t)^m /(t^n (t+1)^p (t+2)^q)\), where \(\hat{H}(t)\) is given by the expression 51 , was computed using Wolfram Mathematica; this required up to 400 seconds CPU time per case for larger \(m\).

5.2 Using an integer relation algorithm to find formulas↩︎

Once a 400-digit value for a given mixed Euler constant was obtained, we employed the multipair PSLQ algorithm to search for integer relations with known constants [11], [16], [17]. Given an \(v\)-long vector \(x = (x_0, x_1, \cdots, x_{v-1})\) of high-precision floating-point reals, the multipair PSLQ algorithm searches for integers \((a_0, a_1, \cdots, a_{v-1})\) such that \(a_0 x_0 + a_1 x_1 + \cdots + a_{v-1} x_{v-1} = 0\) to within available precision, or else establishes that there is no such integer relation within a given bound. The algorithm operates by generating an iterative sequence of \(v \times v\) integer matrices \(B\), so that the entries of the vector \(y = B \cdot x\) become progressively smaller, until one entry of \(y\) is numerically zero, at which iteration the algorithm halts, with the relation given by the row of \(B\) corresponding to the zero entry of \(y\). In the application here, we set \(x_0\) to the 400-digit value of \(M(m,n,p,q)\). For the other entries of the input \(x\) vector, we specified 400-digit values of constants listed in Theorem 1, depending on the order \(r\).

Integer relation detection by any algorithm requires very high precision (at least \(v \cdot \max_i \log_{10} |a_i|\) digits) to produce numerically reliable results, since otherwise the real relation, if any, will be lost in a sea of numerical artifacts. An effective check of numerical reliability with the multipair PSLQ algorithm is to note the dynamic range of the entries of the \(y\) vector at the iteration of detection. In the computer runs for results presented above and in Appendix 8, this dynamic range always exceeded \(10^{63}\), and in most cases exceeded \(10^{300}\). In other words, each of these relations holds to at least 63 digits (and in most cases to more than 300 digits) beyond the level required to discover the relation. However, these results should not be regarded as formally proven by these computations.

Figure 1 illustrates the process of finding a relation using the multipair PSLQ algorithm and assessing the numerical reliability of the result. This shows the base-10 logarithm of the minimum absolute value of the \(y\) vector (vertical axis), plotted against the iteration number (horizontal axis), in the multipair PSLQ computer run that the present authors employed to discover the order-10 formula \[\begin{align} M(1,3,6,0) &= \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{6}} = \frac{ 1}{4}\left( 84\zeta(2) -108\zeta(3) -5\zeta(4) -48\zeta(5) +24\zeta(2)\zeta(3) -9\zeta(6) +6\zeta(3)^2 \right. \nonumber \\ &\left.-12\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4)\right). \label{form:m1360} \end{align}\tag{53}\] Note that as the algorithm proceeds, the minimum absolute value of the \(y\) vector slowly decreases, from approximately \(10^{-3}\) to approximately \(10^{-65}\), but at iteration 311 abruptly drops to approximately \(10^{-405}\), a drop of 340 orders of magnitude. Note that since we are using 400-digit precision, \(10^{-405}\) is effectively zero, so the algorithm terminates here with the relation \((4, -84, 108, 5, 48, -24, 9, -6, 12, -4, -4)\). In other words, formula 53 holds to roughly 340 digits beyond the precision level required to discover it. This dynamic range at the iteration of detection can thus be considered a “confidence level” of the result’s numerical reliability.

5.3 Computational results↩︎

The process described above succeeded in finding relations for each of the cases of form 49 with orders between 3 and 11, a total of 960 cases, plus an additional 54 selected cases of order 12. See Appendix 2 (Section 8) for a complete listing of these formulas. As noted below, in order to minimize the possibility of transcription errors, the LaTeX code for each section of results was generated automatically by a computer program from the output computer files, and this LaTeX code is included here without any alteration.

The techniques described in this section are applicable to more general classes of Euler sums, including Euler sums of orders higher than 12 and Euler sums with more complicated polynomial denominators. However, the computational cost increases with the precision required and the number of selected right-hand-side constants. The principal challenges here are the first and fourth term of 52 , namely (\(a\)) the cost of explicitly computing and summing to high precision a large number of terms of the mixed Euler sum series, and (\(b\)) the symbolic expansion and numerical evaluation of high-order derivatives of the function \(\hat{G}(t)\). Perhaps further investigation into the underlying theory of Euler sums will yield computational schemes that are more efficient for large problems.

6 Conclusions↩︎

We have presented techniques, both algebraic and computational, for finding analytic evaluations of a significantly larger class of Euler sums than studied previously. We believe that most of these formulas are new to the literature. Along this line, we have found that Wolfram Mathematica (version 14.2) can evaluate many of the basic cases, but a large majority are not evaluated by this software.

These methods appear to be applicable to even more general Euler sums. For example, by applying the methods described above, we have obtained these intriguing computational results, among others: \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k(2k+1)} &= 2\log(2)^2 \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(2k+1)} &= 2\zeta(3) -4\log(2)^2 \\ \sum_{k=1}^\infty \frac{H(k)}{(2k+1)^{2}} &= \frac{ 1}{4}\left( 7\zeta(3) -6\log(2)\zeta(2)\right) \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(2k+1)^{2}} &= 9\zeta(3) -6\log(2)\zeta(2) -8\log(2)^2 \\ \sum_{k=1}^\infty \frac{H(k)}{(2k+1)^{4}} &= \frac{ 1}{16}\left( 62\zeta(5) -21\zeta(2)\zeta(3) -30\log(2)\zeta(4)\right) \end{align}\] Note the appearance of \(\log(2)\) in these formulas. Each of these formulas holds to nearly 400-digit accuracy (approximately 350 digits beyond the level required to discover them), but at present we do not yet know how they can be rigorously proven.

The new results presented in this study also highlight the benefits of attempting to solve for all Euler sums of a given order. The results given for sums of mixed type may be of use in indicating the zeta function values likely to arise in attempts to numerically solve for recalcitrant sums like those for order eight and higher. It is hoped that the asymptotic form inferred for the constants \(\gamma_p^H\) can be deduced rigorously, as it may well prove useful in other applications of high-order Euler sums.

7 Appendix 1: 400-digit values of three key constants↩︎

We present here 400-digit approximations of \(M(2,6), \, M(2,8), \, M(3,8), \, M(2,10), \, M(4,8)\):

78483751544719684852509976852158376374740737268847953695380222383595172532123654
63963612795034976112760332996361625685218808108323018034356756036322549570832977
08604139265652530043836463078378465035583569011375448218307043216126923803712749
23988797094981204968396475470138806138535478550733612009250215922048841374239723
645442685850

49180863769095027121905627225975982985135460410529740749826141104503536876835470
18469301862442802589875242849768895787689895958104331283788277223328457927340866
40158920385626435450329285165922784555461987108701748322359094830741802548831985
88668354450902612335818964472409228594433865464246509588184931824643738162119198
662316661058

94671117513355550815252027023563642862142136424802329417381850641867359561102369
07708608852232885420834448581394420559852401108798464519014241848466439384418357
64122407964525143823389069592803884034573487533288088530610292952331243167418134
67198428033583320677784291408584463648948157254030603048103031772735772545074505
256977622009

43078937691748374313286427718378216976785972900132909275418926271525587620211343
23828486038109593077991669275749305259201098402321866062725818826804223344328667
24311160745124744110382924162704634065128094036087399151400598689180216783658166
11238941907596547797317455957463173904843228565114429853394147887119147441919740
167418480233

37811800890914656945480609918216413770587578233999208809687116705691954898386671
55265235208787528310809835281206739252549491701207237871226480178257316430518840
84028017686753324378706573579491021902617968550914371718501262794713100873318573
13123823952255996488552702615920587505801228078818063210192756692513776767867839
561290224768

8 Appendix 2: Formulas for orders 3 through 12↩︎

We present here the full set of results for \(M(m,n,p,q)\) for orders 3 through 11, plus some additional selected cases of order 12. Each of these formulas holds to at least 380-digit precision, which is at least 63 digits (and in most cases more than 300 digits) beyond the level required to discover the relation. However, these formulas should not be regarded as formally proven solely by these computations.

To minimize the possibility of transcription errors, in each section below the formulas were produced by a computer program that parses the computer run output files, extracts the formulas, sorts them lexiographically and then generates LaTeX code (including all spacing, line breaks and page breaks). We then applied a separate program, which parses this LaTeX code, numerically evaluates each of the left- and right-hand sides, and verifies equality to 200-digit precision. No errors were found.

We have included this LaTeX code below without any alteration.

Formulas for order \(r = m + n + p + q = 3\): \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{2}} &= -\left( -2\zeta(3)\right) \tag{54} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)} &= \left( \zeta(2)\right) \tag{55} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{2}} &= -\left( -\zeta(3)\right) \tag{56} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+2)} &= \frac{1}{2}\left( 1 +\zeta(2)\right) \tag{57} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)(k+2)} &= -\left( -1\right) \tag{58} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+2)^{2}} &= \left( -2 +\zeta(2) +\zeta(3)\right) \tag{59} \end{align}\]

Formulas for order \(r = m + n + p + q = 4\): \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{3}} &= \frac{-1}{4}\left( -5\zeta(4)\right) \tag{60} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)} &= -\left( \zeta(2) -2\zeta(3)\right) \tag{61} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{2}} &= \left( \zeta(2) -\zeta(3)\right) \tag{62} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{3}} &= \frac{1}{4}\left( \zeta(4)\right) \tag{63} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+2)} &= \frac{-1}{4}\left( 1 +\zeta(2) -4\zeta(3)\right) \tag{64} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)(k+2)} &= \frac{1}{2}\left( -1 +\zeta(2)\right) \tag{65} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{2}(k+2)} &= -\left( 1 -\zeta(3)\right) \tag{66} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+2)^{2}} &= \frac{1}{4}\left( 5 -\zeta(2) -2\zeta(3)\right) \tag{67} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)(k+2)^{2}} &= -\left( -3 +\zeta(2) +\zeta(3)\right) \tag{68} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+2)^{3}} &= \frac{-1}{4}\left( 12 -4\zeta(2) -4\zeta(3) -\zeta(4)\right) \tag{69} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}} &= \frac{-1}{4}\left( -17\zeta(4)\right) \tag{70} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)} &= \left( 3\zeta(3)\right) \tag{71} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{2}} &= \frac{-1}{4}\left( -11\zeta(4)\right) \tag{72} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+2)} &= \frac{1}{2}\left( 1 +\zeta(2) +3\zeta(3)\right) \tag{73} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)(k+2)} &= -\left( -1 -\zeta(2)\right) \tag{74} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+2)^{2}} &= \frac{-1}{4}\left( 12 -8\zeta(3) -11\zeta(4)\right) \tag{75} \end{align}\]

Formulas for order \(r = m + n + p + q = 5\): \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{4}} &= -\left( -3\zeta(5) +\zeta(2)\zeta(3)\right) \tag{76} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)} &= \frac{-1}{4}\left( -4\zeta(2) +8\zeta(3) -5\zeta(4)\right) \tag{77} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{2}} &= \left( -2\zeta(2) +3\zeta(3)\right) \tag{78} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{3}} &= \frac{1}{4}\left( 4\zeta(2) -4\zeta(3) -\zeta(4)\right) \tag{79} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{4}} &= \left( 2\zeta(5) -\zeta(2)\zeta(3)\right) \tag{80} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+2)} &= \frac{-1}{8}\left( -1 -\zeta(2) +4\zeta(3) -5\zeta(4)\right) \tag{81} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)(k+2)} &= \frac{-1}{4}\left( -1 +3\zeta(2) -4\zeta(3)\right) \tag{82} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{2}(k+2)} &= \frac{1}{2}\left( 1 +\zeta(2) -2\zeta(3)\right) \tag{83} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{3}(k+2)} &= \frac{1}{4}\left( 4 -4\zeta(3) +\zeta(4)\right) \tag{84} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+2)^{2}} &= \frac{1}{4}\left( -3 +3\zeta(3)\right) \tag{85} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)(k+2)^{2}} &= \frac{1}{4}\left( -7 +3\zeta(2) +2\zeta(3)\right) \tag{86} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{2}(k+2)^{2}} &= \left( -4 +\zeta(2) +2\zeta(3)\right) \tag{87} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+2)^{3}} &= \frac{1}{8}\left( 17 -5\zeta(2) -6\zeta(3) -\zeta(4)\right) \tag{88} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)(k+2)^{3}} &= \frac{1}{4}\left( 24 -8\zeta(2) -8\zeta(3) -\zeta(4)\right) \tag{89} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+2)^{4}} &= -\left( 4 -\zeta(2) -\zeta(3) -\zeta(4) -2\zeta(5) +\zeta(2)\zeta(3)\right) \tag{90} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}} &= \frac{-1}{2}\left( -7\zeta(5) +2\zeta(2)\zeta(3)\right) \tag{91} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)} &= \frac{1}{4}\left( -12\zeta(3) +17\zeta(4)\right) \tag{92} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{2}} &= \frac{-1}{4}\left( -12\zeta(3) +11\zeta(4)\right) \tag{93} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{3}} &= \frac{-1}{2}\left( 3\zeta(5) -2\zeta(2)\zeta(3)\right) \tag{94} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+2)} &= \frac{-1}{8}\left( 2 +2\zeta(2) +6\zeta(3) -17\zeta(4)\right) \tag{95} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)(k+2)} &= \frac{1}{2}\left( -1 -\zeta(2) +3\zeta(3)\right) \tag{96} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{2}(k+2)} &= \frac{-1}{4}\left( 4 +4\zeta(2) -11\zeta(4)\right) \tag{97} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+2)^{2}} &= \frac{-1}{8}\left( -14 -2\zeta(2) +2\zeta(3) +11\zeta(4)\right) \tag{98} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)(k+2)^{2}} &= \frac{-1}{4}\left( -16 -4\zeta(2) +8\zeta(3) +11\zeta(4)\right) \tag{99} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+2)^{3}} &= \frac{1}{2}\left( -12 +2\zeta(2) +6\zeta(3) +\zeta(4) -3\zeta(5) +2\zeta(2)\zeta(3)\right) \tag{100} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}} &= -\left( -10\zeta(5) -\zeta(2)\zeta(3)\right) \tag{101} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)} &= -\left( -10\zeta(4)\right) \tag{102} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{2}} &= \frac{-1}{2}\left( -15\zeta(5) -2\zeta(2)\zeta(3)\right) \tag{103} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+2)} &= \frac{-1}{2}\left( -1 -2\zeta(2) -4\zeta(3) -10\zeta(4)\right) \tag{104} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)(k+2)} &= \left( 1 +2\zeta(2) +4\zeta(3)\right) \tag{105} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+2)^{2}} &= \frac{1}{4}\left( -16 -12\zeta(2) -4\zeta(3) +33\zeta(4) +30\zeta(5) +4\zeta(2)\zeta(3)\right) \tag{106} \end{align}\]

Formulas for order \(r = m + n + p + q = 6\): \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{5}} &= \frac{1}{4}\left( 7\zeta(6) -2\zeta(3)^2\right) \tag{107} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)} &= \frac{1}{4}\left( -4\zeta(2) +8\zeta(3) -5\zeta(4) +12\zeta(5) -4\zeta(2)\zeta(3)\right) \tag{108} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{2}} &= \frac{1}{4}\left( 12\zeta(2) -20\zeta(3) +5\zeta(4)\right) \tag{109} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{3}} &= \frac{-1}{4}\left( 12\zeta(2) -16\zeta(3) -\zeta(4)\right) \tag{110} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{4}} &= \frac{1}{4}\left( 4\zeta(2) -4\zeta(3) -\zeta(4) -8\zeta(5) +4\zeta(2)\zeta(3)\right) \tag{111} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{5}} &= \frac{-1}{4}\left( -3\zeta(6) +2\zeta(3)^2\right) \tag{112} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+2)} &= \frac{1}{16}\left( -1 -\zeta(2) +4\zeta(3) -5\zeta(4) +24\zeta(5) -8\zeta(2)\zeta(3)\right) \tag{113} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)(k+2)} &= \frac{-1}{8}\left( 1 -7\zeta(2) +12\zeta(3) -5\zeta(4)\right) \tag{114} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{2}(k+2)} &= \frac{-1}{4}\left( 1 +5\zeta(2) -8\zeta(3)\right) \tag{115} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{3}(k+2)} &= \frac{-1}{4}\left( 2 -2\zeta(2) +\zeta(4)\right) \tag{116} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{4}(k+2)} &= \frac{1}{4}\left( -4 +4\zeta(3) -\zeta(4) +8\zeta(5) -4\zeta(2)\zeta(3)\right) \tag{117} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+2)^{2}} &= \frac{-1}{16}\left( -7 -\zeta(2) +10\zeta(3) -5\zeta(4)\right) \tag{118} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)(k+2)^{2}} &= \frac{-1}{4}\left( -4 +3\zeta(2) -\zeta(3)\right) \tag{119} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{2}(k+2)^{2}} &= \frac{-1}{4}\left( -9 +\zeta(2) +6\zeta(3)\right) \tag{120} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{3}(k+2)^{2}} &= \frac{1}{4}\left( 20 -4\zeta(2) -12\zeta(3) +\zeta(4)\right) \tag{121} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+2)^{3}} &= \frac{-1}{16}\left( 23 -5\zeta(2) -12\zeta(3) -\zeta(4)\right) \tag{122} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k(k+1)(k+2)^{3}} &= \frac{-1}{8}\left( 31 -11\zeta(2) -10\zeta(3) -\zeta(4)\right) \tag{123} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{2}(k+2)^{3}} &= \frac{-1}{4}\left( 40 -12\zeta(2) -16\zeta(3) -\zeta(4)\right) \tag{124} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+2)^{4}} &= \frac{1}{16}\left( 49 -13\zeta(2) -14\zeta(3) -9\zeta(4) -16\zeta(5) +8\zeta(2)\zeta(3)\right) \tag{125} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)(k+2)^{4}} &= \frac{1}{4}\left( 40 -12\zeta(2) -12\zeta(3) -5\zeta(4) -8\zeta(5) +4\zeta(2)\zeta(3)\right) \tag{126} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+2)^{5}} &= \frac{-1}{4}\left( 20 -4\zeta(2) -4\zeta(3) -4\zeta(4) -4\zeta(5) -3\zeta(6) +2\zeta(3)^2\right) \tag{127} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}} &= \frac{1}{24}\left( 97\zeta(6) -48\zeta(3)^2\right) \tag{128} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)} &= \frac{1}{4}\left( 12\zeta(3) -17\zeta(4) +14\zeta(5) -4\zeta(2)\zeta(3)\right) \tag{129} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{2}} &= \left( -6\zeta(3) +7\zeta(4)\right) \tag{130} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{3}} &= \frac{-1}{4}\left( -12\zeta(3) +11\zeta(4) -6\zeta(5) +4\zeta(2)\zeta(3)\right) \tag{131} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{4}} &= \frac{-1}{24}\left( -37\zeta(6) +24\zeta(3)^2\right) \tag{132} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+2)} &= \frac{-1}{16}\left( -2 -2\zeta(2) -6\zeta(3) +17\zeta(4) -28\zeta(5) +8\zeta(2)\zeta(3)\right) \tag{133} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)(k+2)} &= \frac{1}{8}\left( 2 +2\zeta(2) -18\zeta(3) +17\zeta(4)\right) \tag{134} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{2}(k+2)} &= \frac{-1}{4}\left( -2 -2\zeta(2) -6\zeta(3) +11\zeta(4)\right) \tag{135} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{3}(k+2)} &= \frac{1}{4}\left( 4 +4\zeta(2) -11\zeta(4) -6\zeta(5) +4\zeta(2)\zeta(3)\right) \tag{136} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+2)^{2}} &= \frac{1}{4}\left( -4 -\zeta(2) -\zeta(3) +7\zeta(4)\right) \tag{137} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)(k+2)^{2}} &= \frac{1}{8}\left( -18 -6\zeta(2) +14\zeta(3) +11\zeta(4)\right) \tag{138} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{2}(k+2)^{2}} &= \frac{-1}{2}\left( 10 +4\zeta(2) -4\zeta(3) -11\zeta(4)\right) \tag{139} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+2)^{3}} &= \frac{-1}{16}\left( -62 +6\zeta(2) +26\zeta(3) +15\zeta(4) -12\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{140} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)(k+2)^{3}} &= \frac{-1}{4}\left( -40 +20\zeta(3) +13\zeta(4) -6\zeta(5) +4\zeta(2)\zeta(3)\right) \tag{141} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+2)^{4}} &= \frac{1}{24}\left( -240 +48\zeta(2) +96\zeta(3) +36\zeta(4) +96\zeta(5) -48\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+37\zeta(6) -24\zeta(3)^2\right) \tag{142} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}} &= \frac{-1}{16}\left( -93\zeta(6) +40\zeta(3)^2\right) \tag{143} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)} &= \left( -10\zeta(4) +10\zeta(5) +\zeta(2)\zeta(3)\right) \tag{144} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{2}} &= \frac{1}{2}\left( 20\zeta(4) -15\zeta(5) -2\zeta(2)\zeta(3)\right) \tag{145} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{3}} &= \frac{-1}{16}\left( 33\zeta(6) -32\zeta(3)^2\right) \tag{146} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+2)} &= \frac{1}{4}\left( -1 -2\zeta(2) -4\zeta(3) -10\zeta(4) +20\zeta(5) +2\zeta(2)\zeta(3)\right) \tag{147} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)(k+2)} &= \frac{1}{2}\left( -1 -2\zeta(2) -4\zeta(3) +10\zeta(4)\right) \tag{148} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{2}(k+2)} &= \frac{1}{2}\left( -2 -4\zeta(2) -8\zeta(3) +15\zeta(5) +2\zeta(2)\zeta(3)\right) \tag{149} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+2)^{2}} &= \frac{1}{8}\left( 18 +16\zeta(2) +12\zeta(3) -13\zeta(4) -30\zeta(5) -4\zeta(2)\zeta(3)\right) \tag{150} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)(k+2)^{2}} &= \frac{1}{4}\left( 20 +20\zeta(2) +20\zeta(3) -33\zeta(4) -30\zeta(5) \right. \nonumber \\ &\left.-4\zeta(2)\zeta(3)\right) \tag{151} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+2)^{3}} &= \frac{1}{16}\left( -160 -48\zeta(2) +48\zeta(3) +144\zeta(4) -72\zeta(5) +48\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-33\zeta(6) +32\zeta(3)^2\right) \tag{152} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}} &= \frac{1}{24}\left( 979\zeta(6) +72\zeta(3)^2\right) \tag{153} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)} &= \left( 30\zeta(5) +6\zeta(2)\zeta(3)\right) \tag{154} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{2}} &= \frac{-1}{24}\left( -859\zeta(6) -72\zeta(3)^2\right) \tag{155} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+2)} &= \frac{1}{4}\left( 2 +6\zeta(2) +22\zeta(3) +37\zeta(4) +60\zeta(5) +12\zeta(2)\zeta(3)\right) \tag{156} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)(k+2)} &= \frac{-1}{2}\left( -2 -6\zeta(2) -22\zeta(3) -37\zeta(4)\right) \tag{157} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+2)^{2}} &= \frac{-1}{24}\left( 120 +192\zeta(2) +432\zeta(3) +48\zeta(4) -720\zeta(5) -96\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-859\zeta(6) -72\zeta(3)^2\right) \tag{158} \end{align}\]

Formulas for order \(r = m + n + p + q = 7\): \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{6}} &= \left( 4\zeta(7) -\zeta(2)\zeta(5) -\zeta(3)\zeta(4)\right) \tag{159} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)} &= \frac{-1}{4}\left( -4\zeta(2) +8\zeta(3) -5\zeta(4) +12\zeta(5) -4\zeta(2)\zeta(3) -7\zeta(6) \right. \nonumber \\ &\left.+2\zeta(3)^2\right) \tag{160} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{2}} &= \frac{-1}{2}\left( 8\zeta(2) -14\zeta(3) +5\zeta(4) -6\zeta(5) +2\zeta(2)\zeta(3)\right) \tag{161} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{3}} &= -\left( -6\zeta(2) +9\zeta(3) -\zeta(4)\right) \tag{162} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{4}} &= \frac{1}{2}\left( -8\zeta(2) +10\zeta(3) +\zeta(4) +4\zeta(5) -2\zeta(2)\zeta(3)\right) \tag{163} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{5}} &= \frac{-1}{4}\left( -4\zeta(2) +4\zeta(3) +\zeta(4) +8\zeta(5) -4\zeta(2)\zeta(3) +3\zeta(6) \right. \nonumber \\ &\left.-2\zeta(3)^2\right) \tag{164} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{6}} &= -\left( -3\zeta(7) +\zeta(2)\zeta(5) +\zeta(3)\zeta(4)\right) \tag{165} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+2)} &= \frac{1}{32}\left( 1 +\zeta(2) -4\zeta(3) +5\zeta(4) -24\zeta(5) +8\zeta(2)\zeta(3) +28\zeta(6) \right. \nonumber \\ &\left.-8\zeta(3)^2\right) \tag{166} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)(k+2)} &= \frac{1}{16}\left( 1 -15\zeta(2) +28\zeta(3) -15\zeta(4) +24\zeta(5) \right. \nonumber \\ &\left.-8\zeta(2)\zeta(3)\right) \tag{167} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{2}(k+2)} &= \frac{1}{8}\left( 1 +17\zeta(2) -28\zeta(3) +5\zeta(4)\right) \tag{168} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{3}(k+2)} &= \frac{-1}{4}\left( -1 +7\zeta(2) -8\zeta(3) -\zeta(4)\right) \tag{169} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{4}(k+2)} &= \frac{1}{2}\left( 1 +\zeta(2) -2\zeta(3) -4\zeta(5) +2\zeta(2)\zeta(3)\right) \tag{170} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{5}(k+2)} &= \frac{-1}{4}\left( -4 +4\zeta(3) -\zeta(4) +8\zeta(5) -4\zeta(2)\zeta(3) -3\zeta(6) \right. \nonumber \\ &\left.+2\zeta(3)^2\right) \tag{171} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+2)^{2}} &= \frac{-1}{16}\left( 4 +\zeta(2) -7\zeta(3) +5\zeta(4) -12\zeta(5) +4\zeta(2)\zeta(3)\right) \tag{172} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)(k+2)^{2}} &= \frac{1}{16}\left( -9 +13\zeta(2) -14\zeta(3) +5\zeta(4)\right) \tag{173} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{2}(k+2)^{2}} &= \frac{-1}{4}\left( 5 +2\zeta(2) -7\zeta(3)\right) \tag{174} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{3}(k+2)^{2}} &= \frac{1}{4}\left( -11 +3\zeta(2) +6\zeta(3) -\zeta(4)\right) \tag{175} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{4}(k+2)^{2}} &= \frac{1}{2}\left( -12 +2\zeta(2) +8\zeta(3) -\zeta(4) +4\zeta(5) -2\zeta(2)\zeta(3)\right) \tag{176} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+2)^{3}} &= \frac{-1}{16}\left( -15 +2\zeta(2) +11\zeta(3) -2\zeta(4)\right) \tag{177} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)(k+2)^{3}} &= \frac{-1}{16}\left( -39 +17\zeta(2) +8\zeta(3) +\zeta(4)\right) \tag{178} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{2}(k+2)^{3}} &= \frac{-1}{8}\left( -49 +13\zeta(2) +22\zeta(3) +\zeta(4)\right) \tag{179} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{3}(k+2)^{3}} &= \left( 15 -4\zeta(2) -7\zeta(3)\right) \tag{180} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+2)^{4}} &= \frac{1}{16}\left( -36 +9\zeta(2) +13\zeta(3) +5\zeta(4) +8\zeta(5) -4\zeta(2)\zeta(3)\right) \tag{181} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)(k+2)^{4}} &= \frac{1}{16}\left( -111 +35\zeta(2) +34\zeta(3) +11\zeta(4) +16\zeta(5) \right. \nonumber \\ &\left.-8\zeta(2)\zeta(3)\right) \tag{182} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{2}(k+2)^{4}} &= \frac{-1}{2}\left( 40 -12\zeta(2) -14\zeta(3) -3\zeta(4) -4\zeta(5) \right. \nonumber \\ &\left.+2\zeta(2)\zeta(3)\right) \tag{183} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+2)^{5}} &= \frac{-1}{32}\left( -129 +29\zeta(2) +30\zeta(3) +25\zeta(4) +32\zeta(5) -8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+12\zeta(6) -8\zeta(3)^2\right) \tag{184} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)(k+2)^{5}} &= \frac{-1}{4}\left( -60 +16\zeta(2) +16\zeta(3) +9\zeta(4) +12\zeta(5) -4\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+3\zeta(6) -2\zeta(3)^2\right) \tag{185} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+2)^{6}} &= \left( -6 +\zeta(2) +\zeta(3) +\zeta(4) +\zeta(5) +\zeta(6) +3\zeta(7) -\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-\zeta(3)\zeta(4)\right) \tag{186} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}} &= \frac{1}{2}\left( 12\zeta(7) -2\zeta(2)\zeta(5) -5\zeta(3)\zeta(4)\right) \tag{187} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)} &= \frac{-1}{24}\left( 72\zeta(3) -102\zeta(4) +84\zeta(5) -24\zeta(2)\zeta(3) -97\zeta(6) \right. \nonumber \\ &\left.+48\zeta(3)^2\right) \tag{188} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{2}} &= \frac{1}{4}\left( 36\zeta(3) -45\zeta(4) +14\zeta(5) -4\zeta(2)\zeta(3)\right) \tag{189} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{3}} &= \frac{-1}{4}\left( 36\zeta(3) -39\zeta(4) +6\zeta(5) -4\zeta(2)\zeta(3)\right) \tag{190} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{4}} &= \frac{-1}{24}\left( -72\zeta(3) +66\zeta(4) -36\zeta(5) +24\zeta(2)\zeta(3) +37\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2\right) \tag{191} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{5}} &= \frac{1}{2}\left( -2\zeta(7) +2\zeta(2)\zeta(5) -\zeta(3)\zeta(4)\right) \tag{192} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+2)} &= \frac{1}{96}\left( -6 -6\zeta(2) -18\zeta(3) +51\zeta(4) -84\zeta(5) +24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+194\zeta(6) -96\zeta(3)^2\right) \tag{193} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)(k+2)} &= \frac{-1}{16}\left( 2 +2\zeta(2) -42\zeta(3) +51\zeta(4) -28\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{194} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{2}(k+2)} &= \frac{-1}{8}\left( 2 +2\zeta(2) +30\zeta(3) -39\zeta(4)\right) \tag{195} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{3}(k+2)} &= \frac{-1}{2}\left( 1 +\zeta(2) -3\zeta(3) -3\zeta(5) +2\zeta(2)\zeta(3)\right) \tag{196} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{4}(k+2)} &= \frac{-1}{24}\left( 24 +24\zeta(2) -66\zeta(4) -36\zeta(5) +24\zeta(2)\zeta(3) -37\zeta(6) \right. \nonumber \\ &\left.+24\zeta(3)^2\right) \tag{197} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+2)^{2}} &= \frac{-1}{32}\left( -18 -6\zeta(2) -10\zeta(3) +45\zeta(4) -28\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{198} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)(k+2)^{2}} &= \frac{-1}{8}\left( -10 -4\zeta(2) +16\zeta(3) -3\zeta(4)\right) \tag{199} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{2}(k+2)^{2}} &= \frac{1}{8}\left( 22 +10\zeta(2) -2\zeta(3) -33\zeta(4)\right) \tag{200} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{3}(k+2)^{2}} &= \frac{1}{4}\left( 24 +12\zeta(2) -8\zeta(3) -33\zeta(4) -6\zeta(5) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(3)\right) \tag{201} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+2)^{3}} &= \frac{-1}{32}\left( 78 -2\zeta(2) -22\zeta(3) -43\zeta(4) +12\zeta(5) \right. \nonumber \\ &\left.-8\zeta(2)\zeta(3)\right) \tag{202} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)(k+2)^{3}} &= \frac{1}{16}\left( -98 -6\zeta(2) +54\zeta(3) +37\zeta(4) -12\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{203} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{2}(k+2)^{3}} &= \frac{1}{4}\left( -60 -8\zeta(2) +28\zeta(3) +35\zeta(4) -6\zeta(5) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(3)\right) \tag{204} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+2)^{4}} &= \frac{-1}{96}\left( -666 +114\zeta(2) +270\zeta(3) +117\zeta(4) +156\zeta(5) -72\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+74\zeta(6) -48\zeta(3)^2\right) \tag{205} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)(k+2)^{4}} &= \frac{-1}{24}\left( -480 +48\zeta(2) +216\zeta(3) +114\zeta(4) +60\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(3) +37\zeta(6) -24\zeta(3)^2\right) \tag{206} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+2)^{5}} &= \frac{1}{2}\left( -30 +6\zeta(2) +10\zeta(3) +5\zeta(4) +10\zeta(5) -4\zeta(2)\zeta(3) +3\zeta(6) \right. \nonumber \\ &\left.-2\zeta(3)^2 -2\zeta(7) +2\zeta(2)\zeta(5) -\zeta(3)\zeta(4)\right) \tag{207} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}} &= \frac{1}{16}\left( 231\zeta(7) +32\zeta(2)\zeta(5) -204\zeta(3)\zeta(4)\right) \tag{208} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)} &= \frac{-1}{16}\left( -160\zeta(4) +160\zeta(5) +16\zeta(2)\zeta(3) -93\zeta(6) \right. \nonumber \\ &\left.+40\zeta(3)^2\right) \tag{209} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{2}} &= \frac{-1}{2}\left( 40\zeta(4) -35\zeta(5) -4\zeta(2)\zeta(3)\right) \tag{210} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{3}} &= \frac{-1}{16}\left( -160\zeta(4) +120\zeta(5) +16\zeta(2)\zeta(3) -33\zeta(6) \right. \nonumber \\ &\left.+32\zeta(3)^2\right) \tag{211} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{4}} &= \frac{1}{16}\left( 119\zeta(7) +32\zeta(2)\zeta(5) -132\zeta(3)\zeta(4)\right) \tag{212} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+2)} &= \frac{-1}{32}\left( -4 -8\zeta(2) -16\zeta(3) -40\zeta(4) +80\zeta(5) +8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-93\zeta(6) +40\zeta(3)^2\right) \tag{213} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)(k+2)} &= \frac{-1}{4}\left( -1 -2\zeta(2) -4\zeta(3) +30\zeta(4) -20\zeta(5) \right. \nonumber \\ &\left.-2\zeta(2)\zeta(3)\right) \tag{214} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{2}(k+2)} &= \frac{-1}{2}\left( -1 -2\zeta(2) -4\zeta(3) -10\zeta(4) +15\zeta(5) \right. \nonumber \\ &\left.+2\zeta(2)\zeta(3)\right) \tag{215} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{3}(k+2)} &= \frac{1}{16}\left( 16 +32\zeta(2) +64\zeta(3) -120\zeta(5) -16\zeta(2)\zeta(3) -33\zeta(6) \right. \nonumber \\ &\left.+32\zeta(3)^2\right) \tag{216} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+2)^{2}} &= \frac{-1}{16}\left( 20 +20\zeta(2) +20\zeta(3) +7\zeta(4) -70\zeta(5) \right. \nonumber \\ &\left.-8\zeta(2)\zeta(3)\right) \tag{217} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)(k+2)^{2}} &= \frac{-1}{8}\left( 22 +24\zeta(2) +28\zeta(3) -53\zeta(4) -30\zeta(5) \right. \nonumber \\ &\left.-4\zeta(2)\zeta(3)\right) \tag{218} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{2}(k+2)^{2}} &= \frac{1}{4}\left( -24 -28\zeta(2) -36\zeta(3) +33\zeta(4) +60\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{219} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+2)^{3}} &= \frac{-1}{32}\left( -196 -80\zeta(2) +24\zeta(3) +170\zeta(4) -12\zeta(5) +56\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-33\zeta(6) +32\zeta(3)^2\right) \tag{220} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)(k+2)^{3}} &= \frac{1}{16}\left( 240 +128\zeta(2) +32\zeta(3) -276\zeta(4) -48\zeta(5) \right. \nonumber \\ &\left.-64\zeta(2)\zeta(3) +33\zeta(6) -32\zeta(3)^2\right) \tag{221} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+2)^{4}} &= \frac{1}{16}\left( -320 -32\zeta(2) +128\zeta(3) +172\zeta(4) +24\zeta(5) +74\zeta(6) \right. \nonumber \\ &\left.-48\zeta(3)^2 +119\zeta(7) +32\zeta(2)\zeta(5) -132\zeta(3)\zeta(4)\right) \tag{222} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}} &= \frac{-1}{8}\left( -185\zeta(7) -40\zeta(2)\zeta(5) +172\zeta(3)\zeta(4)\right) \tag{223} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)} &= \frac{1}{24}\left( -720\zeta(5) -144\zeta(2)\zeta(3) +979\zeta(6) +72\zeta(3)^2\right) \tag{224} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{2}} &= \frac{1}{24}\left( 720\zeta(5) +144\zeta(2)\zeta(3) -859\zeta(6) -72\zeta(3)^2\right) \tag{225} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{3}} &= \frac{1}{8}\left( -109\zeta(7) -40\zeta(2)\zeta(5) +148\zeta(3)\zeta(4)\right) \tag{226} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+2)} &= \frac{-1}{48}\left( 12 +36\zeta(2) +132\zeta(3) +222\zeta(4) +360\zeta(5) +72\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-979\zeta(6) -72\zeta(3)^2\right) \tag{227} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)(k+2)} &= \frac{-1}{4}\left( 2 +6\zeta(2) +22\zeta(3) +37\zeta(4) -60\zeta(5) \right. \nonumber \\ &\left.-12\zeta(2)\zeta(3)\right) \tag{228} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{2}(k+2)} &= \frac{1}{24}\left( -24 -72\zeta(2) -264\zeta(3) -444\zeta(4) +859\zeta(6) \right. \nonumber \\ &\left.+72\zeta(3)^2\right) \tag{229} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+2)^{2}} &= \frac{1}{48}\left( 132 +228\zeta(2) +564\zeta(3) +270\zeta(4) -360\zeta(5) -24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-859\zeta(6) -72\zeta(3)^2\right) \tag{230} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)(k+2)^{2}} &= \frac{1}{24}\left( 144 +264\zeta(2) +696\zeta(3) +492\zeta(4) -720\zeta(5) \right. \nonumber \\ &\left.-96\zeta(2)\zeta(3) -859\zeta(6) -72\zeta(3)^2\right) \tag{231} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+2)^{3}} &= \frac{-1}{8}\left( 120 +112\zeta(2) +160\zeta(3) -124\zeta(4) -168\zeta(5) -80\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+66\zeta(6) -64\zeta(3)^2 +109\zeta(7) +40\zeta(2)\zeta(5) -148\zeta(3)\zeta(4)\right) \tag{232} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}} &= \frac{1}{16}\left( 2051\zeta(7) +456\zeta(2)\zeta(5) +528\zeta(3)\zeta(4)\right) \tag{233} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)} &= \frac{-1}{2}\left( -357\zeta(6) -45\zeta(3)^2\right) \tag{234} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{2}} &= \frac{-1}{16}\left( -1855\zeta(7) -456\zeta(2)\zeta(5) -528\zeta(3)\zeta(4)\right) \tag{235} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+2)} &= \frac{-1}{8}\left( -4 -16\zeta(2) -84\zeta(3) -251\zeta(4) -284\zeta(5) -60\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-714\zeta(6) -90\zeta(3)^2\right) \tag{236} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)(k+2)} &= \frac{1}{4}\left( 4 +16\zeta(2) +84\zeta(3) +251\zeta(4) +284\zeta(5) \right. \nonumber \\ &\left.+60\zeta(2)\zeta(3)\right) \tag{237} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+2)^{2}} &= \frac{-1}{48}\left( 288 +720\zeta(2) +2784\zeta(3) +4704\zeta(4) -192\zeta(5) \right. \nonumber \\ &\left.+240\zeta(2)\zeta(3) -8590\zeta(6) -720\zeta(3)^2 -5565\zeta(7) -1368\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1584\zeta(3)\zeta(4)\right) \tag{238} \end{align}\]

Formulas for order \(r = m + n + p + q = 8\): \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{7}} &= \frac{1}{4}\left( 9\zeta(8) -4\zeta(3)\zeta(5)\right) \tag{239} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+1)} &= \frac{1}{4}\left( -4\zeta(2) +8\zeta(3) -5\zeta(4) +12\zeta(5) -4\zeta(2)\zeta(3) -7\zeta(6) \right. \nonumber \\ &\left.+2\zeta(3)^2 +16\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4)\right) \tag{240} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)^{2}} &= \frac{1}{4}\left( 20\zeta(2) -36\zeta(3) +15\zeta(4) -24\zeta(5) +8\zeta(2)\zeta(3) +7\zeta(6) \right. \nonumber \\ &\left.-2\zeta(3)^2\right) \tag{241} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{3}} &= \frac{1}{2}\left( -20\zeta(2) +32\zeta(3) -7\zeta(4) +6\zeta(5) -2\zeta(2)\zeta(3)\right) \tag{242} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{4}} &= \frac{1}{2}\left( 20\zeta(2) -28\zeta(3) +\zeta(4) -4\zeta(5) +2\zeta(2)\zeta(3)\right) \tag{243} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{5}} &= \frac{1}{4}\left( -20\zeta(2) +24\zeta(3) +3\zeta(4) +16\zeta(5) -8\zeta(2)\zeta(3) +3\zeta(6) \right. \nonumber \\ &\left.-2\zeta(3)^2\right) \tag{244} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{6}} &= \frac{-1}{4}\left( -4\zeta(2) +4\zeta(3) +\zeta(4) +8\zeta(5) -4\zeta(2)\zeta(3) +3\zeta(6) \right. \nonumber \\ &\left.-2\zeta(3)^2 +12\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4)\right) \tag{245} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{7}} &= \frac{-1}{4}\left( -5\zeta(8) +4\zeta(3)\zeta(5)\right) \tag{246} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+2)} &= \frac{-1}{64}\left( 1 +\zeta(2) -4\zeta(3) +5\zeta(4) -24\zeta(5) +8\zeta(2)\zeta(3) +28\zeta(6) \right. \nonumber \\ &\left.-8\zeta(3)^2 -128\zeta(7) +32\zeta(2)\zeta(5) +32\zeta(3)\zeta(4)\right) \tag{247} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)(k+2)} &= \frac{-1}{32}\left( 1 -31\zeta(2) +60\zeta(3) -35\zeta(4) +72\zeta(5) -24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-28\zeta(6) +8\zeta(3)^2\right) \tag{248} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{2}(k+2)} &= \frac{-1}{16}\left( 1 +49\zeta(2) -84\zeta(3) +25\zeta(4) -24\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{249} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{3}(k+2)} &= \frac{-1}{8}\left( 1 -31\zeta(2) +44\zeta(3) -3\zeta(4)\right) \tag{250} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{4}(k+2)} &= \frac{-1}{4}\left( 1 +9\zeta(2) -12\zeta(3) -\zeta(4) -8\zeta(5) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(3)\right) \tag{251} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{5}(k+2)} &= \frac{1}{4}\left( -2 +2\zeta(2) -\zeta(4) -3\zeta(6) +2\zeta(3)^2\right) \tag{252} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{6}(k+2)} &= \frac{1}{4}\left( -4 +4\zeta(3) -\zeta(4) +8\zeta(5) -4\zeta(2)\zeta(3) -3\zeta(6) +2\zeta(3)^2 \right. \nonumber \\ &\left.+12\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4)\right) \tag{253} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+2)^{2}} &= \frac{-1}{64}\left( -9 -3\zeta(2) +18\zeta(3) -15\zeta(4) +48\zeta(5) -16\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-28\zeta(6) +8\zeta(3)^2\right) \tag{254} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)(k+2)^{2}} &= \frac{-1}{16}\left( -5 +14\zeta(2) -21\zeta(3) +10\zeta(4) -12\zeta(5) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(3)\right) \tag{255} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{2}(k+2)^{2}} &= \frac{1}{16}\left( 11 +21\zeta(2) -42\zeta(3) +5\zeta(4)\right) \tag{256} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{3}(k+2)^{2}} &= \frac{1}{4}\left( 6 -5\zeta(2) +\zeta(3) +\zeta(4)\right) \tag{257} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{4}(k+2)^{2}} &= \frac{1}{4}\left( 13 -\zeta(2) -10\zeta(3) +\zeta(4) -8\zeta(5) +4\zeta(2)\zeta(3)\right) \tag{258} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{5}(k+2)^{2}} &= \frac{-1}{4}\left( -28 +4\zeta(2) +20\zeta(3) -3\zeta(4) +16\zeta(5) -8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-3\zeta(6) +2\zeta(3)^2\right) \tag{259} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+2)^{3}} &= \frac{1}{32}\left( -19 +\zeta(2) +18\zeta(3) -7\zeta(4) +12\zeta(5) -4\zeta(2)\zeta(3)\right) \tag{260} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)(k+2)^{3}} &= \frac{-1}{16}\left( 24 -15\zeta(2) +3\zeta(3) -3\zeta(4)\right) \tag{261} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{2}(k+2)^{3}} &= \frac{-1}{16}\left( 59 -9\zeta(2) -36\zeta(3) -\zeta(4)\right) \tag{262} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{3}(k+2)^{3}} &= \frac{-1}{8}\left( 71 -19\zeta(2) -34\zeta(3) +\zeta(4)\right) \tag{263} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{4}(k+2)^{3}} &= \frac{1}{2}\left( -42 +10\zeta(2) +22\zeta(3) -\zeta(4) +4\zeta(5) \right. \nonumber \\ &\left.-2\zeta(2)\zeta(3)\right) \tag{264} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+2)^{4}} &= \frac{1}{32}\left( 51 -11\zeta(2) -24\zeta(3) -3\zeta(4) -8\zeta(5) +4\zeta(2)\zeta(3)\right) \tag{265} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)(k+2)^{4}} &= \frac{-1}{16}\left( -75 +26\zeta(2) +21\zeta(3) +6\zeta(4) +8\zeta(5) \right. \nonumber \\ &\left.-4\zeta(2)\zeta(3)\right) \tag{266} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{2}(k+2)^{4}} &= \frac{1}{16}\left( 209 -61\zeta(2) -78\zeta(3) -13\zeta(4) -16\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{267} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{3}(k+2)^{4}} &= \frac{1}{2}\left( 70 -20\zeta(2) -28\zeta(3) -3\zeta(4) -4\zeta(5) \right. \nonumber \\ &\left.+2\zeta(2)\zeta(3)\right) \tag{268} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+2)^{5}} &= \frac{-1}{64}\left( 201 -47\zeta(2) -56\zeta(3) -35\zeta(4) -48\zeta(5) +16\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-12\zeta(6) +8\zeta(3)^2\right) \tag{269} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)(k+2)^{5}} &= \frac{1}{32}\left( -351 +99\zeta(2) +98\zeta(3) +47\zeta(4) +64\zeta(5) -24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+12\zeta(6) -8\zeta(3)^2\right) \tag{270} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{2}(k+2)^{5}} &= \frac{1}{4}\left( -140 +40\zeta(2) +44\zeta(3) +15\zeta(4) +20\zeta(5) -8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+3\zeta(6) -2\zeta(3)^2\right) \tag{271} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+2)^{6}} &= \frac{1}{64}\left( 321 -61\zeta(2) -62\zeta(3) -57\zeta(4) -64\zeta(5) +8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-44\zeta(6) +8\zeta(3)^2 -96\zeta(7) +32\zeta(2)\zeta(5) +32\zeta(3)\zeta(4)\right) \tag{272} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)(k+2)^{6}} &= \frac{1}{4}\left( 84 -20\zeta(2) -20\zeta(3) -13\zeta(4) -16\zeta(5) +4\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-7\zeta(6) +2\zeta(3)^2 -12\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4)\right) \tag{273} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+2)^{7}} &= \frac{1}{4}\left( -28 +4\zeta(2) +4\zeta(3) +4\zeta(4) +4\zeta(5) +4\zeta(6) +4\zeta(7) +5\zeta(8) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(5)\right) \tag{274} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{6}} &= -\left( - M(2,6)\right) \tag{275} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+1)} &= \frac{-1}{24}\left( -72\zeta(3) +102\zeta(4) -84\zeta(5) +24\zeta(2)\zeta(3) +97\zeta(6) \right. \nonumber \\ &\left.-48\zeta(3)^2 -144\zeta(7) +24\zeta(2)\zeta(5) +60\zeta(3)\zeta(4)\right) \tag{276} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)^{2}} &= \frac{-1}{24}\left( 288\zeta(3) -372\zeta(4) +168\zeta(5) -48\zeta(2)\zeta(3) -97\zeta(6) \right. \nonumber \\ &\left.+48\zeta(3)^2\right) \tag{277} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{3}} &= \left( 18\zeta(3) -21\zeta(4) +5\zeta(5) -2\zeta(2)\zeta(3)\right) \tag{278} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{4}} &= \frac{-1}{24}\left( 288\zeta(3) -300\zeta(4) +72\zeta(5) -48\zeta(2)\zeta(3) -37\zeta(6) \right. \nonumber \\ &\left.+24\zeta(3)^2\right) \tag{279} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{5}} &= \frac{1}{24}\left( 72\zeta(3) -66\zeta(4) +36\zeta(5) -24\zeta(2)\zeta(3) -37\zeta(6) \right. \nonumber \\ &\left.+24\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) +12\zeta(3)\zeta(4)\right) \tag{280} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{6}} &= \frac{1}{2}\left( -7\zeta(8) +4\zeta(3)\zeta(5) +2 M(2,6)\right) \tag{281} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+2)} &= \frac{1}{192}\left( 6 +6\zeta(2) +18\zeta(3) -51\zeta(4) +84\zeta(5) -24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-194\zeta(6) +96\zeta(3)^2 +576\zeta(7) -96\zeta(2)\zeta(5) -240\zeta(3)\zeta(4)\right) \tag{282} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)(k+2)} &= \frac{-1}{96}\left( -6 -6\zeta(2) +270\zeta(3) -357\zeta(4) +252\zeta(5) \right. \nonumber \\ &\left.-72\zeta(2)\zeta(3) -194\zeta(6) +96\zeta(3)^2\right) \tag{283} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{2}(k+2)} &= \frac{-1}{16}\left( -2 -2\zeta(2) -102\zeta(3) +129\zeta(4) -28\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{284} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{3}(k+2)} &= \frac{1}{8}\left( 2 +2\zeta(2) -42\zeta(3) +39\zeta(4) -12\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{285} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{4}(k+2)} &= \frac{-1}{24}\left( -12 -12\zeta(2) -36\zeta(3) +66\zeta(4) +37\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2\right) \tag{286} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{5}(k+2)} &= \frac{-1}{24}\left( -24 -24\zeta(2) +66\zeta(4) +36\zeta(5) -24\zeta(2)\zeta(3) +37\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) +12\zeta(3)\zeta(4)\right) \tag{287} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+2)^{2}} &= \frac{-1}{96}\left( 30 +12\zeta(2) +24\zeta(3) -93\zeta(4) +84\zeta(5) -24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-97\zeta(6) +48\zeta(3)^2\right) \tag{288} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)(k+2)^{2}} &= \frac{-1}{32}\left( 22 +10\zeta(2) -74\zeta(3) +57\zeta(4) -28\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{289} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{2}(k+2)^{2}} &= \frac{-1}{4}\left( 6 +3\zeta(2) +7\zeta(3) -18\zeta(4)\right) \tag{290} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{3}(k+2)^{2}} &= \frac{1}{8}\left( -26 -14\zeta(2) +14\zeta(3) +33\zeta(4) +12\zeta(5) \right. \nonumber \\ &\left.-8\zeta(2)\zeta(3)\right) \tag{291} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{4}(k+2)^{2}} &= \frac{-1}{24}\left( 168 +96\zeta(2) -48\zeta(3) -264\zeta(4) -72\zeta(5) \right. \nonumber \\ &\left.+48\zeta(2)\zeta(3) -37\zeta(6) +24\zeta(3)^2\right) \tag{292} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+2)^{3}} &= \frac{-1}{16}\left( -24 -\zeta(2) +3\zeta(3) +22\zeta(4) -10\zeta(5) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(3)\right) \tag{293} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)(k+2)^{3}} &= \frac{-1}{32}\left( -118 -14\zeta(2) +86\zeta(3) +31\zeta(4) -12\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{294} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{2}(k+2)^{3}} &= \frac{1}{16}\left( 142 +26\zeta(2) -58\zeta(3) -103\zeta(4) +12\zeta(5) \right. \nonumber \\ &\left.-8\zeta(2)\zeta(3)\right) \tag{295} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{3}(k+2)^{3}} &= \left( 21 +5\zeta(2) -9\zeta(3) -17\zeta(4)\right) \tag{296} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+2)^{4}} &= \frac{1}{96}\left( -450 +60\zeta(2) +168\zeta(3) +123\zeta(4) +60\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(3) +37\zeta(6) -24\zeta(3)^2\right) \tag{297} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)(k+2)^{4}} &= \frac{1}{96}\left( -1254 +78\zeta(2) +594\zeta(3) +339\zeta(4) +84\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(3) +74\zeta(6) -48\zeta(3)^2\right) \tag{298} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{2}(k+2)^{4}} &= \frac{-1}{24}\left( 840 -384\zeta(3) -324\zeta(4) -24\zeta(5) -37\zeta(6) \right. \nonumber \\ &\left.+24\zeta(3)^2\right) \tag{299} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+2)^{5}} &= \frac{1}{192}\left( 2106 -402\zeta(2) -750\zeta(3) -357\zeta(4) -636\zeta(5) \right. \nonumber \\ &\left.+264\zeta(2)\zeta(3) -218\zeta(6) +144\zeta(3)^2 +96\zeta(7) -96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+48\zeta(3)\zeta(4)\right) \tag{300} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)(k+2)^{5}} &= \frac{1}{24}\left( 840 -120\zeta(2) -336\zeta(3) -174\zeta(4) -180\zeta(5) \right. \nonumber \\ &\left.+72\zeta(2)\zeta(3) -73\zeta(6) +48\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+12\zeta(3)\zeta(4)\right) \tag{301} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+2)^{6}} &= \frac{1}{2}\left( -42 +8\zeta(2) +12\zeta(3) +7\zeta(4) +12\zeta(5) -4\zeta(2)\zeta(3) +5\zeta(6) \right. \nonumber \\ &\left.-2\zeta(3)^2 +12\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4) -7\zeta(8) +4\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+2 M(2,6)\right) \tag{302} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{5}} &= \frac{1}{96}\left( -595\zeta(8) -120\zeta(2)\zeta(3)^2 +576\zeta(3)\zeta(5) +264 M(2,6)\right) \tag{303} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+1)} &= \frac{-1}{16}\left( 160\zeta(4) -160\zeta(5) -16\zeta(2)\zeta(3) +93\zeta(6) -40\zeta(3)^2 \right. \nonumber \\ &\left.-231\zeta(7) -32\zeta(2)\zeta(5) +204\zeta(3)\zeta(4)\right) \tag{304} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)^{2}} &= \frac{1}{16}\left( 480\zeta(4) -440\zeta(5) -48\zeta(2)\zeta(3) +93\zeta(6) \right. \nonumber \\ &\left.-40\zeta(3)^2\right) \tag{305} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{3}} &= \frac{-1}{16}\left( 480\zeta(4) -400\zeta(5) -48\zeta(2)\zeta(3) +33\zeta(6) \right. \nonumber \\ &\left.-32\zeta(3)^2\right) \tag{306} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{4}} &= \frac{-1}{16}\left( -160\zeta(4) +120\zeta(5) +16\zeta(2)\zeta(3) -33\zeta(6) +32\zeta(3)^2 \right. \nonumber \\ &\left.+119\zeta(7) +32\zeta(2)\zeta(5) -132\zeta(3)\zeta(4)\right) \tag{307} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{5}} &= \frac{1}{96}\left( -43\zeta(8) -120\zeta(2)\zeta(3)^2 +288\zeta(3)\zeta(5) -24 M(2,6)\right) \tag{308} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+2)} &= \frac{-1}{64}\left( 4 +8\zeta(2) +16\zeta(3) +40\zeta(4) -80\zeta(5) -8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+93\zeta(6) -40\zeta(3)^2 -462\zeta(7) -64\zeta(2)\zeta(5) +408\zeta(3)\zeta(4)\right) \tag{309} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)(k+2)} &= \frac{-1}{32}\left( 4 +8\zeta(2) +16\zeta(3) -280\zeta(4) +240\zeta(5) +24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-93\zeta(6) +40\zeta(3)^2\right) \tag{310} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{2}(k+2)} &= \frac{1}{4}\left( -1 -2\zeta(2) -4\zeta(3) -50\zeta(4) +50\zeta(5) \right. \nonumber \\ &\left.+6\zeta(2)\zeta(3)\right) \tag{311} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{3}(k+2)} &= \frac{1}{16}\left( -8 -16\zeta(2) -32\zeta(3) +80\zeta(4) +33\zeta(6) -32\zeta(3)^2\right) \tag{312} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{4}(k+2)} &= \frac{-1}{16}\left( 16 +32\zeta(2) +64\zeta(3) -120\zeta(5) -16\zeta(2)\zeta(3) -33\zeta(6) \right. \nonumber \\ &\left.+32\zeta(3)^2 -119\zeta(7) -32\zeta(2)\zeta(5) +132\zeta(3)\zeta(4)\right) \tag{313} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+2)^{2}} &= \frac{-1}{64}\left( -44 -48\zeta(2) -56\zeta(3) -54\zeta(4) +220\zeta(5) +24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-93\zeta(6) +40\zeta(3)^2\right) \tag{314} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)(k+2)^{2}} &= \frac{-1}{16}\left( -24 -28\zeta(2) -36\zeta(3) +113\zeta(4) -10\zeta(5)\right) \tag{315} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{2}(k+2)^{2}} &= \frac{-1}{8}\left( -26 -32\zeta(2) -44\zeta(3) +13\zeta(4) +90\zeta(5) \right. \nonumber \\ &\left.+12\zeta(2)\zeta(3)\right) \tag{316} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{3}(k+2)^{2}} &= \frac{1}{16}\left( 112 +144\zeta(2) +208\zeta(3) -132\zeta(4) -360\zeta(5) \right. \nonumber \\ &\left.-48\zeta(2)\zeta(3) -33\zeta(6) +32\zeta(3)^2\right) \tag{317} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+2)^{3}} &= \frac{1}{64}\left( -236 -120\zeta(2) -16\zeta(3) +156\zeta(4) +128\zeta(5) \right. \nonumber \\ &\left.+72\zeta(2)\zeta(3) -33\zeta(6) +32\zeta(3)^2\right) \tag{318} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)(k+2)^{3}} &= \frac{-1}{32}\left( 284 +176\zeta(2) +88\zeta(3) -382\zeta(4) -108\zeta(5) \right. \nonumber \\ &\left.-72\zeta(2)\zeta(3) +33\zeta(6) -32\zeta(3)^2\right) \tag{319} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{2}(k+2)^{3}} &= \frac{-1}{16}\left( 336 +240\zeta(2) +176\zeta(3) -408\zeta(4) -288\zeta(5) \right. \nonumber \\ &\left.-96\zeta(2)\zeta(3) +33\zeta(6) -32\zeta(3)^2\right) \tag{320} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+2)^{4}} &= \frac{-1}{64}\left( -836 -144\zeta(2) +280\zeta(3) +514\zeta(4) +36\zeta(5) +56\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+115\zeta(6) -64\zeta(3)^2 +238\zeta(7) +64\zeta(2)\zeta(5) -264\zeta(3)\zeta(4)\right) \tag{321} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)(k+2)^{4}} &= \frac{-1}{16}\left( -560 -160\zeta(2) +96\zeta(3) +448\zeta(4) +72\zeta(5) \right. \nonumber \\ &\left.+64\zeta(2)\zeta(3) +41\zeta(6) -16\zeta(3)^2 +119\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-132\zeta(3)\zeta(4)\right) \tag{322} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+2)^{5}} &= \frac{-1}{96}\left( 3360 -1344\zeta(3) -1296\zeta(4) -816\zeta(5) +288\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-660\zeta(6) +432\zeta(3)^2 +288\zeta(7) -288\zeta(2)\zeta(5) +144\zeta(3)\zeta(4) +43\zeta(8) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3)^2 -288\zeta(3)\zeta(5) +24 M(2,6)\right) \tag{323} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{4}} &= \frac{-1}{144}\left( 14833\zeta(8) +4032\zeta(2)\zeta(3)^2 -16704\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-3744 M(2,6)\right) \tag{324} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+1)} &= \frac{1}{24}\left( 720\zeta(5) +144\zeta(2)\zeta(3) -979\zeta(6) -72\zeta(3)^2 +555\zeta(7) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(5) -516\zeta(3)\zeta(4)\right) \tag{325} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)^{2}} &= \frac{1}{12}\left( -720\zeta(5) -144\zeta(2)\zeta(3) +919\zeta(6) +72\zeta(3)^2\right) \tag{326} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{3}} &= \frac{1}{24}\left( 720\zeta(5) +144\zeta(2)\zeta(3) -859\zeta(6) -72\zeta(3)^2 +327\zeta(7) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(5) -444\zeta(3)\zeta(4)\right) \tag{327} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{4}} &= \frac{1}{144}\left( -12415\zeta(8) -3312\zeta(2)\zeta(3)^2 +13824\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+3024 M(2,6)\right) \tag{328} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+2)} &= \frac{1}{96}\left( 12 +36\zeta(2) +132\zeta(3) +222\zeta(4) +360\zeta(5) +72\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-979\zeta(6) -72\zeta(3)^2 +1110\zeta(7) +240\zeta(2)\zeta(5) -1032\zeta(3)\zeta(4)\right) \tag{329} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)(k+2)} &= \frac{1}{48}\left( 12 +36\zeta(2) +132\zeta(3) +222\zeta(4) -1080\zeta(5) \right. \nonumber \\ &\left.-216\zeta(2)\zeta(3) +979\zeta(6) +72\zeta(3)^2\right) \tag{330} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{2}(k+2)} &= \frac{-1}{24}\left( -12 -36\zeta(2) -132\zeta(3) -222\zeta(4) -360\zeta(5) \right. \nonumber \\ &\left.-72\zeta(2)\zeta(3) +859\zeta(6) +72\zeta(3)^2\right) \tag{331} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{3}(k+2)} &= \frac{-1}{24}\left( -24 -72\zeta(2) -264\zeta(3) -444\zeta(4) +859\zeta(6) +72\zeta(3)^2 \right. \nonumber \\ &\left.+327\zeta(7) +120\zeta(2)\zeta(5) -444\zeta(3)\zeta(4)\right) \tag{332} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+2)^{2}} &= \frac{1}{48}\left( -72 -132\zeta(2) -348\zeta(3) -246\zeta(4) -24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+919\zeta(6) +72\zeta(3)^2\right) \tag{333} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)(k+2)^{2}} &= \frac{-1}{48}\left( 156 +300\zeta(2) +828\zeta(3) +714\zeta(4) -1080\zeta(5) \right. \nonumber \\ &\left.-168\zeta(2)\zeta(3) -859\zeta(6) -72\zeta(3)^2\right) \tag{334} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{2}(k+2)^{2}} &= \frac{1}{12}\left( -84 -168\zeta(2) -480\zeta(3) -468\zeta(4) +360\zeta(5) \right. \nonumber \\ &\left.+48\zeta(2)\zeta(3) +859\zeta(6) +72\zeta(3)^2\right) \tag{335} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+2)^{3}} &= \frac{1}{96}\left( 852 +900\zeta(2) +1524\zeta(3) -474\zeta(4) -1368\zeta(5) \right. \nonumber \\ &\left.-504\zeta(2)\zeta(3) -463\zeta(6) -456\zeta(3)^2 +654\zeta(7) +240\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-888\zeta(3)\zeta(4)\right) \tag{336} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)(k+2)^{3}} &= \frac{-1}{24}\left( -504 -600\zeta(2) -1176\zeta(3) -120\zeta(4) +1224\zeta(5) \right. \nonumber \\ &\left.+336\zeta(2)\zeta(3) +661\zeta(6) +264\zeta(3)^2 -327\zeta(7) -120\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+444\zeta(3)\zeta(4)\right) \tag{337} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+2)^{4}} &= \frac{-1}{144}\left( 5040 +2880\zeta(2) +2304\zeta(3) -5040\zeta(4) -2880\zeta(5) \right. \nonumber \\ &\left.-1728\zeta(2)\zeta(3) -144\zeta(6) -288\zeta(3)^2 -4284\zeta(7) -1152\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+4752\zeta(3)\zeta(4) +12415\zeta(8) +3312\zeta(2)\zeta(3)^2 -13824\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-3024 M(2,6)\right) \tag{338} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{3}} &= \frac{1}{288}\left( -67811\zeta(8) -19080\zeta(2)\zeta(3)^2 +78768\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+16920 M(2,6)\right) \tag{339} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+1)} &= \frac{-1}{16}\left( 2856\zeta(6) +360\zeta(3)^2 -2051\zeta(7) -456\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-528\zeta(3)\zeta(4)\right) \tag{340} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)^{2}} &= \frac{-1}{16}\left( -2856\zeta(6) -360\zeta(3)^2 +1855\zeta(7) +456\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+528\zeta(3)\zeta(4)\right) \tag{341} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{3}} &= \frac{-1}{288}\left( -65621\zeta(8) -17640\zeta(2)\zeta(3)^2 +72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+15480 M(2,6)\right) \tag{342} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+2)} &= \frac{-1}{32}\left( 8 +32\zeta(2) +168\zeta(3) +502\zeta(4) +568\zeta(5) +120\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+1428\zeta(6) +180\zeta(3)^2 -2051\zeta(7) -456\zeta(2)\zeta(5) -528\zeta(3)\zeta(4)\right) \tag{343} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)(k+2)} &= \frac{1}{8}\left( -4 -16\zeta(2) -84\zeta(3) -251\zeta(4) -284\zeta(5) -60\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+714\zeta(6) +90\zeta(3)^2\right) \tag{344} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{2}(k+2)} &= \frac{1}{16}\left( -16 -64\zeta(2) -336\zeta(3) -1004\zeta(4) -1136\zeta(5) \right. \nonumber \\ &\left.-240\zeta(2)\zeta(3) +1855\zeta(7) +456\zeta(2)\zeta(5) +528\zeta(3)\zeta(4)\right) \tag{345} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+2)^{2}} &= \frac{1}{96}\left( 312 +816\zeta(2) +3288\zeta(3) +6210\zeta(4) +1512\zeta(5) \right. \nonumber \\ &\left.+600\zeta(2)\zeta(3) -4306\zeta(6) -180\zeta(3)^2 -5565\zeta(7) -1368\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1584\zeta(3)\zeta(4)\right) \tag{346} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)(k+2)^{2}} &= \frac{-1}{48}\left( -336 -912\zeta(2) -3792\zeta(3) -7716\zeta(4) -3216\zeta(5) \right. \nonumber \\ &\left.-960\zeta(2)\zeta(3) +8590\zeta(6) +720\zeta(3)^2 +5565\zeta(7) +1368\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+1584\zeta(3)\zeta(4)\right) \tag{347} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+2)^{3}} &= \frac{-1}{288}\left( 6048 +10080\zeta(2) +30240\zeta(3) +30096\zeta(4) -18432\zeta(5) \right. \nonumber \\ &\left.-4320\zeta(2)\zeta(3) -45600\zeta(6) -10080\zeta(3)^2 +19620\zeta(7) +7200\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-26640\zeta(3)\zeta(4) -65621\zeta(8) -17640\zeta(2)\zeta(3)^2 +72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+15480 M(2,6)\right) \tag{348} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{2}} &= \frac{-1}{8}\left( -5843\zeta(8) +328\zeta(2)\zeta(3)^2 -3896\zeta(3)\zeta(5) -456 M(2,6)\right) \tag{349} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+1)} &= -\left( -644\zeta(7) -145\zeta(2)\zeta(5) -297\zeta(3)\zeta(4)\right) \tag{350} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)^{2}} &= \frac{-1}{24}\left( -17027\zeta(8) +924\zeta(2)\zeta(3)^2 -11328\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-1308 M(2,6)\right) \tag{351} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+2)} &= \frac{1}{8}\left( 4 +20\zeta(2) +136\zeta(3) +571\zeta(4) +1142\zeta(5) +244\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+2097\zeta(6) +268\zeta(3)^2 +2576\zeta(7) +580\zeta(2)\zeta(5) +1188\zeta(3)\zeta(4)\right) \tag{352} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)(k+2)} &= \frac{1}{4}\left( 4 +20\zeta(2) +136\zeta(3) +571\zeta(4) +1142\zeta(5) +244\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+2097\zeta(6) +268\zeta(3)^2\right) \tag{353} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+2)^{2}} &= \frac{-1}{24}\left( 168 +576\zeta(2) +3120\zeta(3) +9288\zeta(4) +10104\zeta(5) \right. \nonumber \\ &\left.+2448\zeta(2)\zeta(3) -303\zeta(6) +528\zeta(3)^2 -16695\zeta(7) -4104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-4752\zeta(3)\zeta(4) -17027\zeta(8) +924\zeta(2)\zeta(3)^2 -11328\zeta(3)\zeta(5) -1308 M(2,6)\right) \label{eq08116} \end{align}\tag{354}\]

Formulas for order \(r = m + n + p + q = 9\): \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{8}} &= -\left( -5\zeta(9) +\zeta(3)\zeta(6) +\zeta(4)\zeta(5) +\zeta(2)\zeta(7)\right) \tag{355} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{7}(k+1)} &= \frac{-1}{4}\left( -4\zeta(2) +8\zeta(3) -5\zeta(4) +12\zeta(5) -4\zeta(2)\zeta(3) -7\zeta(6) \right. \nonumber \\ &\left.+2\zeta(3)^2 +16\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4) -9\zeta(8) +4\zeta(3)\zeta(5)\right) \tag{356} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+1)^{2}} &= \frac{-1}{2}\left( 12\zeta(2) -22\zeta(3) +10\zeta(4) -18\zeta(5) +6\zeta(2)\zeta(3) +7\zeta(6) \right. \nonumber \\ &\left.-2\zeta(3)^2 -8\zeta(7) +2\zeta(2)\zeta(5) +2\zeta(3)\zeta(4)\right) \tag{357} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)^{3}} &= \frac{1}{4}\left( 60\zeta(2) -100\zeta(3) +29\zeta(4) -36\zeta(5) +12\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+7\zeta(6) -2\zeta(3)^2\right) \tag{358} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{4}} &= \left( -20\zeta(2) +30\zeta(3) -4\zeta(4) +5\zeta(5) -2\zeta(2)\zeta(3)\right) \tag{359} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{5}} &= \frac{1}{4}\left( 60\zeta(2) -80\zeta(3) -\zeta(4) -24\zeta(5) +12\zeta(2)\zeta(3) -3\zeta(6) \right. \nonumber \\ &\left.+2\zeta(3)^2\right) \tag{360} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{6}} &= \frac{-1}{2}\left( 12\zeta(2) -14\zeta(3) -2\zeta(4) -12\zeta(5) +6\zeta(2)\zeta(3) -3\zeta(6) \right. \nonumber \\ &\left.+2\zeta(3)^2 -6\zeta(7) +2\zeta(2)\zeta(5) +2\zeta(3)\zeta(4)\right) \tag{361} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{7}} &= \frac{-1}{4}\left( -4\zeta(2) +4\zeta(3) +\zeta(4) +8\zeta(5) -4\zeta(2)\zeta(3) +3\zeta(6) \right. \nonumber \\ &\left.-2\zeta(3)^2 +12\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4) +5\zeta(8) -4\zeta(3)\zeta(5)\right) \tag{362} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{8}} &= \left( 4\zeta(9) -\zeta(3)\zeta(6) -\zeta(4)\zeta(5) -\zeta(2)\zeta(7)\right) \tag{363} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{7}(k+2)} &= \frac{1}{128}\left( 1 +\zeta(2) -4\zeta(3) +5\zeta(4) -24\zeta(5) +8\zeta(2)\zeta(3) +28\zeta(6) \right. \nonumber \\ &\left.-8\zeta(3)^2 -128\zeta(7) +32\zeta(2)\zeta(5) +32\zeta(3)\zeta(4) +144\zeta(8) \right. \nonumber \\ &\left.-64\zeta(3)\zeta(5)\right) \tag{364} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+1)(k+2)} &= \frac{-1}{64}\left( -1 +63\zeta(2) -124\zeta(3) +75\zeta(4) -168\zeta(5) +56\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+84\zeta(6) -24\zeta(3)^2 -128\zeta(7) +32\zeta(2)\zeta(5) +32\zeta(3)\zeta(4)\right) \tag{365} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)^{2}(k+2)} &= \frac{-1}{32}\left( -1 -129\zeta(2) +228\zeta(3) -85\zeta(4) +120\zeta(5) \right. \nonumber \\ &\left.-40\zeta(2)\zeta(3) -28\zeta(6) +8\zeta(3)^2\right) \tag{366} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{3}(k+2)} &= \frac{1}{16}\left( 1 -111\zeta(2) +172\zeta(3) -31\zeta(4) +24\zeta(5) \right. \nonumber \\ &\left.-8\zeta(2)\zeta(3)\right) \tag{367} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{4}(k+2)} &= \frac{-1}{8}\left( -1 -49\zeta(2) +68\zeta(3) -\zeta(4) +16\zeta(5) \right. \nonumber \\ &\left.-8\zeta(2)\zeta(3)\right) \tag{368} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{5}(k+2)} &= \frac{1}{4}\left( 1 -11\zeta(2) +12\zeta(3) +2\zeta(4) +8\zeta(5) -4\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+3\zeta(6) -2\zeta(3)^2\right) \tag{369} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{6}(k+2)} &= \frac{1}{2}\left( 1 +\zeta(2) -2\zeta(3) -4\zeta(5) +2\zeta(2)\zeta(3) -6\zeta(7) \right. \nonumber \\ &\left.+2\zeta(2)\zeta(5) +2\zeta(3)\zeta(4)\right) \tag{370} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{7}(k+2)} &= \frac{-1}{4}\left( -4 +4\zeta(3) -\zeta(4) +8\zeta(5) -4\zeta(2)\zeta(3) -3\zeta(6) +2\zeta(3)^2 \right. \nonumber \\ &\left.+12\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4) -5\zeta(8) +4\zeta(3)\zeta(5)\right) \tag{371} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+2)^{2}} &= \frac{1}{64}\left( -5 -2\zeta(2) +11\zeta(3) -10\zeta(4) +36\zeta(5) -12\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-28\zeta(6) +8\zeta(3)^2 +64\zeta(7) -16\zeta(2)\zeta(5) -16\zeta(3)\zeta(4)\right) \tag{372} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)(k+2)^{2}} &= \frac{1}{64}\left( -11 +59\zeta(2) -102\zeta(3) +55\zeta(4) -96\zeta(5) +32\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+28\zeta(6) -8\zeta(3)^2\right) \tag{373} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{2}(k+2)^{2}} &= \frac{-1}{16}\left( 6 +35\zeta(2) -63\zeta(3) +15\zeta(4) -12\zeta(5) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(3)\right) \tag{374} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{3}(k+2)^{2}} &= \frac{1}{16}\left( -13 +41\zeta(2) -46\zeta(3) +\zeta(4)\right) \tag{375} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{4}(k+2)^{2}} &= \frac{-1}{4}\left( 7 +4\zeta(2) -11\zeta(3) -8\zeta(5) +4\zeta(2)\zeta(3)\right) \tag{376} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{5}(k+2)^{2}} &= \frac{1}{4}\left( -15 +3\zeta(2) +10\zeta(3) -2\zeta(4) +8\zeta(5) -4\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-3\zeta(6) +2\zeta(3)^2\right) \tag{377} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{6}(k+2)^{2}} &= \frac{1}{2}\left( -16 +2\zeta(2) +12\zeta(3) -2\zeta(4) +12\zeta(5) -6\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-3\zeta(6) +2\zeta(3)^2 +6\zeta(7) -2\zeta(2)\zeta(5) -2\zeta(3)\zeta(4)\right) \tag{378} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+2)^{3}} &= \frac{-1}{128}\left( -47 -\zeta(2) +54\zeta(3) -29\zeta(4) +72\zeta(5) -24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-28\zeta(6) +8\zeta(3)^2\right) \tag{379} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)(k+2)^{3}} &= \frac{-1}{32}\left( -29 +29\zeta(2) -24\zeta(3) +13\zeta(4) -12\zeta(5) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(3)\right) \tag{380} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{2}(k+2)^{3}} &= \frac{-1}{16}\left( -35 -6\zeta(2) +39\zeta(3) -2\zeta(4)\right) \tag{381} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{3}(k+2)^{3}} &= \frac{-1}{16}\left( -83 +29\zeta(2) +32\zeta(3) -3\zeta(4)\right) \tag{382} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{4}(k+2)^{3}} &= \frac{-1}{8}\left( -97 +21\zeta(2) +54\zeta(3) -3\zeta(4) +16\zeta(5) \right. \nonumber \\ &\left.-8\zeta(2)\zeta(3)\right) \tag{383} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{5}(k+2)^{3}} &= \frac{-1}{4}\left( -112 +24\zeta(2) +64\zeta(3) -5\zeta(4) +24\zeta(5) -12\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-3\zeta(6) +2\zeta(3)^2\right) \tag{384} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+2)^{4}} &= \frac{-1}{32}\left( 35 -6\zeta(2) -21\zeta(3) +2\zeta(4) -10\zeta(5) +4\zeta(2)\zeta(3)\right) \tag{385} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)(k+2)^{4}} &= \frac{-1}{32}\left( 99 -41\zeta(2) -18\zeta(3) -9\zeta(4) -8\zeta(5) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(3)\right) \tag{386} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{2}(k+2)^{4}} &= \frac{-1}{16}\left( 134 -35\zeta(2) -57\zeta(3) -7\zeta(4) -8\zeta(5) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(3)\right) \tag{387} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{3}(k+2)^{4}} &= \frac{-1}{16}\left( 351 -99\zeta(2) -146\zeta(3) -11\zeta(4) -16\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{388} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{4}(k+2)^{4}} &= -\left( 56 -15\zeta(2) -25\zeta(3) -\zeta(4) -4\zeta(5) +2\zeta(2)\zeta(3)\right) \tag{389} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+2)^{5}} &= \frac{-1}{128}\left( -303 +69\zeta(2) +104\zeta(3) +41\zeta(4) +64\zeta(5) -24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+12\zeta(6) -8\zeta(3)^2\right) \tag{390} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)(k+2)^{5}} &= \frac{-1}{64}\left( -501 +151\zeta(2) +140\zeta(3) +59\zeta(4) +80\zeta(5) \right. \nonumber \\ &\left.-32\zeta(2)\zeta(3) +12\zeta(6) -8\zeta(3)^2\right) \tag{391} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{2}(k+2)^{5}} &= \frac{1}{32}\left( 769 -221\zeta(2) -254\zeta(3) -73\zeta(4) -96\zeta(5) \right. \nonumber \\ &\left.+40\zeta(2)\zeta(3) -12\zeta(6) +8\zeta(3)^2\right) \tag{392} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{3}(k+2)^{5}} &= \frac{1}{4}\left( 280 -80\zeta(2) -100\zeta(3) -21\zeta(4) -28\zeta(5) +12\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-3\zeta(6) +2\zeta(3)^2\right) \tag{393} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+2)^{6}} &= \frac{1}{64}\left( -261 +54\zeta(2) +59\zeta(3) +46\zeta(4) +56\zeta(5) -12\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+28\zeta(6) -8\zeta(3)^2 +48\zeta(7) -16\zeta(2)\zeta(5) -16\zeta(3)\zeta(4)\right) \tag{394} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)(k+2)^{6}} &= \frac{-1}{64}\left( 1023 -259\zeta(2) -258\zeta(3) -151\zeta(4) -192\zeta(5) \right. \nonumber \\ &\left.+56\zeta(2)\zeta(3) -68\zeta(6) +24\zeta(3)^2 -96\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+32\zeta(3)\zeta(4)\right) \tag{395} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{2}(k+2)^{6}} &= \frac{1}{2}\left( -112 +30\zeta(2) +32\zeta(3) +14\zeta(4) +18\zeta(5) -6\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+5\zeta(6) -2\zeta(3)^2 +6\zeta(7) -2\zeta(2)\zeta(5) -2\zeta(3)\zeta(4)\right) \tag{396} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+2)^{7}} &= \frac{1}{128}\left( 769 -125\zeta(2) -126\zeta(3) -121\zeta(4) -128\zeta(5) +8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-108\zeta(6) +8\zeta(3)^2 -160\zeta(7) +32\zeta(2)\zeta(5) +32\zeta(3)\zeta(4) -80\zeta(8) \right. \nonumber \\ &\left.+64\zeta(3)\zeta(5)\right) \tag{397} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)(k+2)^{7}} &= \frac{-1}{4}\left( -112 +24\zeta(2) +24\zeta(3) +17\zeta(4) +20\zeta(5) -4\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+11\zeta(6) -2\zeta(3)^2 +16\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4) +5\zeta(8) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(5)\right) \tag{398} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+2)^{8}} &= -\left( 8 -\zeta(2) -\zeta(3) -\zeta(4) -\zeta(5) -\zeta(6) -\zeta(7) -\zeta(8) -4\zeta(9) \right. \nonumber \\ &\left.+\zeta(3)\zeta(6) +\zeta(4)\zeta(5) +\zeta(2)\zeta(7)\right) \tag{399} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{7}} &= \frac{-1}{6}\left( -55\zeta(9) +21\zeta(3)\zeta(6) +15\zeta(4)\zeta(5) +6\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-2\zeta(3)^3\right) \tag{400} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{6}(k+1)} &= \frac{1}{24}\left( -72\zeta(3) +102\zeta(4) -84\zeta(5) +24\zeta(2)\zeta(3) +97\zeta(6) \right. \nonumber \\ &\left.-48\zeta(3)^2 -144\zeta(7) +24\zeta(2)\zeta(5) +60\zeta(3)\zeta(4) +24 M(2,6)\right) \tag{401} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+1)^{2}} &= \frac{-1}{12}\left( -180\zeta(3) +237\zeta(4) -126\zeta(5) +36\zeta(2)\zeta(3) +97\zeta(6) \right. \nonumber \\ &\left.-48\zeta(3)^2 -72\zeta(7) +12\zeta(2)\zeta(5) +30\zeta(3)\zeta(4)\right) \tag{402} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)^{3}} &= \frac{1}{24}\left( -720\zeta(3) +876\zeta(4) -288\zeta(5) +96\zeta(2)\zeta(3) +97\zeta(6) \right. \nonumber \\ &\left.-48\zeta(3)^2\right) \tag{403} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{4}} &= \frac{-1}{24}\left( -720\zeta(3) +804\zeta(4) -192\zeta(5) +96\zeta(2)\zeta(3) +37\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2\right) \tag{404} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{5}} &= \frac{1}{12}\left( -180\zeta(3) +183\zeta(4) -54\zeta(5) +36\zeta(2)\zeta(3) +37\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2 -12\zeta(7) +12\zeta(2)\zeta(5) -6\zeta(3)\zeta(4)\right) \tag{405} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{6}} &= \frac{1}{24}\left( 72\zeta(3) -66\zeta(4) +36\zeta(5) -24\zeta(2)\zeta(3) -37\zeta(6) \right. \nonumber \\ &\left.+24\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) +12\zeta(3)\zeta(4) +84\zeta(8) -48\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-24 M(2,6)\right) \tag{406} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{7}} &= \frac{-1}{6}\left( -\zeta(9) +9\zeta(3)\zeta(6) +3\zeta(4)\zeta(5) -6\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-2\zeta(3)^3\right) \tag{407} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{6}(k+2)} &= \frac{1}{384}\left( -6 -6\zeta(2) -18\zeta(3) +51\zeta(4) -84\zeta(5) +24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+194\zeta(6) -96\zeta(3)^2 -576\zeta(7) +96\zeta(2)\zeta(5) +240\zeta(3)\zeta(4) +192 M(2,6)\right) \tag{408} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+1)(k+2)} &= \frac{-1}{64}\left( 2 +2\zeta(2) -186\zeta(3) +255\zeta(4) -196\zeta(5) \right. \nonumber \\ &\left.+56\zeta(2)\zeta(3) +194\zeta(6) -96\zeta(3)^2 -192\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+80\zeta(3)\zeta(4)\right) \tag{409} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)^{2}(k+2)} &= \frac{1}{96}\left( -6 -6\zeta(2) -882\zeta(3) +1131\zeta(4) -420\zeta(5) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3) +194\zeta(6) -96\zeta(3)^2\right) \tag{410} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{3}(k+2)} &= \frac{-1}{16}\left( 2 +2\zeta(2) -186\zeta(3) +207\zeta(4) -52\zeta(5) \right. \nonumber \\ &\left.+24\zeta(2)\zeta(3)\right) \tag{411} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{4}(k+2)} &= \frac{-1}{24}\left( 6 +6\zeta(2) +162\zeta(3) -183\zeta(4) +36\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(3) -37\zeta(6) +24\zeta(3)^2\right) \tag{412} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{5}(k+2)} &= \frac{-1}{2}\left( 1 +\zeta(2) -3\zeta(3) -3\zeta(5) +2\zeta(2)\zeta(3) -2\zeta(7) \right. \nonumber \\ &\left.+2\zeta(2)\zeta(5) -\zeta(3)\zeta(4)\right) \tag{413} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{6}(k+2)} &= \frac{1}{24}\left( -24 -24\zeta(2) +66\zeta(4) +36\zeta(5) -24\zeta(2)\zeta(3) +37\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) +12\zeta(3)\zeta(4) -84\zeta(8) +48\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+24 M(2,6)\right) \tag{414} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+2)^{2}} &= \frac{1}{384}\left( 66 +30\zeta(2) +66\zeta(3) -237\zeta(4) +252\zeta(5) \right. \nonumber \\ &\left.-72\zeta(2)\zeta(3) -388\zeta(6) +192\zeta(3)^2 +576\zeta(7) -96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-240\zeta(3)\zeta(4)\right) \tag{415} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)(k+2)^{2}} &= \frac{1}{96}\left( 36 +18\zeta(2) -246\zeta(3) +264\zeta(4) -168\zeta(5) \right. \nonumber \\ &\left.+48\zeta(2)\zeta(3) +97\zeta(6) -48\zeta(3)^2\right) \tag{416} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{2}(k+2)^{2}} &= \frac{1}{32}\left( 26 +14\zeta(2) +130\zeta(3) -201\zeta(4) +28\zeta(5) \right. \nonumber \\ &\left.-8\zeta(2)\zeta(3)\right) \tag{417} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{3}(k+2)^{2}} &= \frac{1}{8}\left( 14 +8\zeta(2) -28\zeta(3) +3\zeta(4) -12\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3)\right) \tag{418} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{4}(k+2)^{2}} &= \frac{1}{24}\left( 90 +54\zeta(2) -6\zeta(3) -165\zeta(4) -36\zeta(5) \right. \nonumber \\ &\left.+24\zeta(2)\zeta(3) -37\zeta(6) +24\zeta(3)^2\right) \tag{419} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{5}(k+2)^{2}} &= \frac{-1}{12}\left( -96 -60\zeta(2) +24\zeta(3) +165\zeta(4) +54\zeta(5) \right. \nonumber \\ &\left.-36\zeta(2)\zeta(3) +37\zeta(6) -24\zeta(3)^2 +12\zeta(7) -12\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+6\zeta(3)\zeta(4)\right) \tag{420} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+2)^{3}} &= \frac{-1}{192}\left( 174 +18\zeta(2) +6\zeta(3) -225\zeta(4) +144\zeta(5) \right. \nonumber \\ &\left.-48\zeta(2)\zeta(3) -97\zeta(6) +48\zeta(3)^2\right) \tag{421} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)(k+2)^{3}} &= \frac{-1}{32}\left( 70 +12\zeta(2) -80\zeta(3) +13\zeta(4) -8\zeta(5)\right) \tag{422} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{2}(k+2)^{3}} &= \frac{-1}{32}\left( 166 +38\zeta(2) -30\zeta(3) -175\zeta(4) +12\zeta(5) \right. \nonumber \\ &\left.-8\zeta(2)\zeta(3)\right) \tag{423} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{3}(k+2)^{3}} &= \frac{1}{16}\left( -194 -54\zeta(2) +86\zeta(3) +169\zeta(4) +12\zeta(5) \right. \nonumber \\ &\left.-8\zeta(2)\zeta(3)\right) \tag{424} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{4}(k+2)^{3}} &= \frac{-1}{24}\left( 672 +216\zeta(2) -264\zeta(3) -672\zeta(4) -72\zeta(5) \right. \nonumber \\ &\left.+48\zeta(2)\zeta(3) -37\zeta(6) +24\zeta(3)^2\right) \tag{425} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+2)^{4}} &= \frac{1}{192}\left( 594 -54\zeta(2) -186\zeta(3) -255\zeta(4) -37\zeta(6) \right. \nonumber \\ &\left.+24\zeta(3)^2\right) \tag{426} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)(k+2)^{4}} &= \frac{-1}{96}\left( -804 +18\zeta(2) +426\zeta(3) +216\zeta(4) +24\zeta(5) +37\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2\right) \tag{427} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{2}(k+2)^{4}} &= \frac{-1}{96}\left( -2106 -78\zeta(2) +942\zeta(3) +957\zeta(4) +12\zeta(5) \right. \nonumber \\ &\left.+24\zeta(2)\zeta(3) +74\zeta(6) -48\zeta(3)^2\right) \tag{428} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{3}(k+2)^{4}} &= \frac{-1}{24}\left( -1344 -120\zeta(2) +600\zeta(3) +732\zeta(4) +24\zeta(5) +37\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2\right) \tag{429} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+2)^{5}} &= \frac{-1}{384}\left( 3006 -522\zeta(2) -1086\zeta(3) -603\zeta(4) -756\zeta(5) \right. \nonumber \\ &\left.+312\zeta(2)\zeta(3) -292\zeta(6) +192\zeta(3)^2 +96\zeta(7) -96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+48\zeta(3)\zeta(4)\right) \tag{430} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)(k+2)^{5}} &= \frac{1}{64}\left( -1538 +186\zeta(2) +646\zeta(3) +345\zeta(4) +268\zeta(5) \right. \nonumber \\ &\left.-104\zeta(2)\zeta(3) +122\zeta(6) -80\zeta(3)^2 -32\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-16\zeta(3)\zeta(4)\right) \tag{431} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{2}(k+2)^{5}} &= \frac{1}{12}\left( -840 +60\zeta(2) +360\zeta(3) +249\zeta(4) +102\zeta(5) \right. \nonumber \\ &\left.-36\zeta(2)\zeta(3) +55\zeta(6) -36\zeta(3)^2 -12\zeta(7) +12\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-6\zeta(3)\zeta(4)\right) \tag{432} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+2)^{6}} &= \frac{1}{384}\left( 6138 -1170\zeta(2) -1902\zeta(3) -1029\zeta(4) -1788\zeta(5) \right. \nonumber \\ &\left.+648\zeta(2)\zeta(3) -698\zeta(6) +336\zeta(3)^2 -1056\zeta(7) +288\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+432\zeta(3)\zeta(4) +672\zeta(8) -384\zeta(3)\zeta(5) -192 M(2,6)\right) \tag{433} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)(k+2)^{6}} &= \frac{-1}{24}\left( -1344 +216\zeta(2) +480\zeta(3) +258\zeta(4) +324\zeta(5) \right. \nonumber \\ &\left.-120\zeta(2)\zeta(3) +133\zeta(6) -72\zeta(3)^2 +120\zeta(7) -24\zeta(2)\zeta(5) -60\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-84\zeta(8) +48\zeta(3)\zeta(5) +24 M(2,6)\right) \tag{434} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+2)^{7}} &= \frac{1}{6}\left( -168 +30\zeta(2) +42\zeta(3) +27\zeta(4) +42\zeta(5) -12\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+21\zeta(6) -6\zeta(3)^2 +42\zeta(7) -12\zeta(2)\zeta(5) -12\zeta(3)\zeta(4) +15\zeta(8) \right. \nonumber \\ &\left.-12\zeta(3)\zeta(5) +\zeta(9) -9\zeta(3)\zeta(6) -3\zeta(4)\zeta(5) +6\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+2\zeta(3)^3\right) \tag{435} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{6}} &= \frac{1}{24}\left( 521\zeta(9) -291\zeta(3)\zeta(6) -306\zeta(4)\zeta(5) +72\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+48\zeta(3)^3\right) \tag{436} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{5}(k+1)} &= \frac{1}{96}\left( 960\zeta(4) -960\zeta(5) -96\zeta(2)\zeta(3) +558\zeta(6) -240\zeta(3)^2 \right. \nonumber \\ &\left.-1386\zeta(7) -192\zeta(2)\zeta(5) +1224\zeta(3)\zeta(4) -595\zeta(8) -120\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+576\zeta(3)\zeta(5) +264 M(2,6)\right) \tag{437} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+1)^{2}} &= \frac{1}{16}\left( -640\zeta(4) +600\zeta(5) +64\zeta(2)\zeta(3) -186\zeta(6) +80\zeta(3)^2 \right. \nonumber \\ &\left.+231\zeta(7) +32\zeta(2)\zeta(5) -204\zeta(3)\zeta(4)\right) \tag{438} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)^{3}} &= \frac{1}{8}\left( 480\zeta(4) -420\zeta(5) -48\zeta(2)\zeta(3) +63\zeta(6) \right. \nonumber \\ &\left.-36\zeta(3)^2\right) \tag{439} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{4}} &= \frac{1}{16}\left( -640\zeta(4) +520\zeta(5) +64\zeta(2)\zeta(3) -66\zeta(6) +64\zeta(3)^2 \right. \nonumber \\ &\left.+119\zeta(7) +32\zeta(2)\zeta(5) -132\zeta(3)\zeta(4)\right) \tag{440} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{5}} &= \frac{1}{96}\left( 960\zeta(4) -720\zeta(5) -96\zeta(2)\zeta(3) +198\zeta(6) -192\zeta(3)^2 \right. \nonumber \\ &\left.-714\zeta(7) -192\zeta(2)\zeta(5) +792\zeta(3)\zeta(4) +43\zeta(8) +120\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-288\zeta(3)\zeta(5) +24 M(2,6)\right) \tag{441} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{6}} &= \frac{-1}{24}\left( -197\zeta(9) +111\zeta(3)\zeta(6) +198\zeta(4)\zeta(5) -72\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-24\zeta(3)^3\right) \tag{442} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{5}(k+2)} &= \frac{-1}{384}\left( -12 -24\zeta(2) -48\zeta(3) -120\zeta(4) +240\zeta(5) +24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-279\zeta(6) +120\zeta(3)^2 +1386\zeta(7) +192\zeta(2)\zeta(5) -1224\zeta(3)\zeta(4) +1190\zeta(8) \right. \nonumber \\ &\left.+240\zeta(2)\zeta(3)^2 -1152\zeta(3)\zeta(5) -528 M(2,6)\right) \tag{443} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+1)(k+2)} &= \frac{1}{64}\left( 4 +8\zeta(2) +16\zeta(3) -600\zeta(4) +560\zeta(5) +56\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-279\zeta(6) +120\zeta(3)^2 +462\zeta(7) +64\zeta(2)\zeta(5) -408\zeta(3)\zeta(4)\right) \tag{444} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)^{2}(k+2)} &= \frac{1}{32}\left( 4 +8\zeta(2) +16\zeta(3) +680\zeta(4) -640\zeta(5) \right. \nonumber \\ &\left.-72\zeta(2)\zeta(3) +93\zeta(6) -40\zeta(3)^2\right) \tag{445} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{3}(k+2)} &= \frac{1}{16}\left( 4 +8\zeta(2) +16\zeta(3) -280\zeta(4) +200\zeta(5) \right. \nonumber \\ &\left.+24\zeta(2)\zeta(3) -33\zeta(6) +32\zeta(3)^2\right) \tag{446} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{4}(k+2)} &= \frac{-1}{16}\left( -8 -16\zeta(2) -32\zeta(3) -80\zeta(4) +120\zeta(5) +16\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+119\zeta(7) +32\zeta(2)\zeta(5) -132\zeta(3)\zeta(4)\right) \tag{447} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{5}(k+2)} &= \frac{1}{96}\left( 96 +192\zeta(2) +384\zeta(3) -720\zeta(5) -96\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-198\zeta(6) +192\zeta(3)^2 -714\zeta(7) -192\zeta(2)\zeta(5) +792\zeta(3)\zeta(4) -43\zeta(8) \right. \nonumber \\ &\left.-120\zeta(2)\zeta(3)^2 +288\zeta(3)\zeta(5) -24 M(2,6)\right) \tag{448} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+2)^{2}} &= \frac{-1}{64}\left( 24 +28\zeta(2) +36\zeta(3) +47\zeta(4) -150\zeta(5) -16\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+93\zeta(6) -40\zeta(3)^2 -231\zeta(7) -32\zeta(2)\zeta(5) +204\zeta(3)\zeta(4)\right) \tag{449} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)(k+2)^{2}} &= \frac{-1}{64}\left( 52 +64\zeta(2) +88\zeta(3) -506\zeta(4) +260\zeta(5) \right. \nonumber \\ &\left.+24\zeta(2)\zeta(3) -93\zeta(6) +40\zeta(3)^2\right) \tag{450} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{2}(k+2)^{2}} &= \frac{1}{16}\left( -28 -36\zeta(2) -52\zeta(3) -87\zeta(4) +190\zeta(5) \right. \nonumber \\ &\left.+24\zeta(2)\zeta(3)\right) \tag{451} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{3}(k+2)^{2}} &= \frac{1}{16}\left( -60 -80\zeta(2) -120\zeta(3) +106\zeta(4) +180\zeta(5) \right. \nonumber \\ &\left.+24\zeta(2)\zeta(3) +33\zeta(6) -32\zeta(3)^2\right) \tag{452} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{4}(k+2)^{2}} &= \frac{-1}{16}\left( 128 +176\zeta(2) +272\zeta(3) -132\zeta(4) -480\zeta(5) \right. \nonumber \\ &\left.-64\zeta(2)\zeta(3) -66\zeta(6) +64\zeta(3)^2 -119\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+132\zeta(3)\zeta(4)\right) \tag{453} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+2)^{3}} &= \frac{-1}{64}\left( -140 -84\zeta(2) -36\zeta(3) +51\zeta(4) +174\zeta(5) \right. \nonumber \\ &\left.+48\zeta(2)\zeta(3) -63\zeta(6) +36\zeta(3)^2\right) \tag{454} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)(k+2)^{3}} &= \frac{-1}{64}\left( -332 -232\zeta(2) -160\zeta(3) +608\zeta(4) +88\zeta(5) \right. \nonumber \\ &\left.+72\zeta(2)\zeta(3) -33\zeta(6) +32\zeta(3)^2\right) \tag{455} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{2}(k+2)^{3}} &= \frac{1}{32}\left( 388 +304\zeta(2) +264\zeta(3) -434\zeta(4) -468\zeta(5) \right. \nonumber \\ &\left.-120\zeta(2)\zeta(3) +33\zeta(6) -32\zeta(3)^2\right) \tag{456} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{3}(k+2)^{3}} &= \frac{1}{4}\left( 112 +96\zeta(2) +96\zeta(3) -135\zeta(4) -162\zeta(5) \right. \nonumber \\ &\left.-36\zeta(2)\zeta(3)\right) \tag{457} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+2)^{4}} &= \frac{-1}{64}\left( 536 +132\zeta(2) -132\zeta(3) -335\zeta(4) -82\zeta(5) \right. \nonumber \\ &\left.-64\zeta(2)\zeta(3) -41\zeta(6) +16\zeta(3)^2 -119\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+132\zeta(3)\zeta(4)\right) \tag{458} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)(k+2)^{4}} &= \frac{-1}{64}\left( 1404 +496\zeta(2) -104\zeta(3) -1278\zeta(4) -252\zeta(5) \right. \nonumber \\ &\left.-200\zeta(2)\zeta(3) -49\zeta(6) -238\zeta(7) -64\zeta(2)\zeta(5) +264\zeta(3)\zeta(4)\right) \tag{459} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{2}(k+2)^{4}} &= \frac{1}{16}\left( -896 -400\zeta(2) -80\zeta(3) +856\zeta(4) +360\zeta(5) \right. \nonumber \\ &\left.+160\zeta(2)\zeta(3) +8\zeta(6) +16\zeta(3)^2 +119\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-132\zeta(3)\zeta(4)\right) \tag{460} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+2)^{5}} &= \frac{1}{384}\left( 9228 +432\zeta(2) -3528\zeta(3) -4134\zeta(4) -1740\zeta(5) \right. \nonumber \\ &\left.+408\zeta(2)\zeta(3) -1665\zeta(6) +1056\zeta(3)^2 -138\zeta(7) -768\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+1080\zeta(3)\zeta(4) +86\zeta(8) +240\zeta(2)\zeta(3)^2 -576\zeta(3)\zeta(5) +48 M(2,6)\right) \tag{461} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)(k+2)^{5}} &= \frac{-1}{96}\left( -6720 -960\zeta(2) +1920\zeta(3) +3984\zeta(4) +1248\zeta(5) \right. \nonumber \\ &\left.+96\zeta(2)\zeta(3) +906\zeta(6) -528\zeta(3)^2 +426\zeta(7) +480\zeta(2)\zeta(5) -936\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-43\zeta(8) -120\zeta(2)\zeta(3)^2 +288\zeta(3)\zeta(5) -24 M(2,6)\right) \tag{462} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+2)^{6}} &= \frac{1}{24}\left( -1344 +72\zeta(2) +504\zeta(3) +414\zeta(4) +396\zeta(5) -144\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+243\zeta(6) -144\zeta(3)^2 +144\zeta(7) -108\zeta(3)\zeta(4) -252\zeta(8) +144\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+72 M(2,6) +197\zeta(9) -111\zeta(3)\zeta(6) -198\zeta(4)\zeta(5) +72\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+24\zeta(3)^3\right) \tag{463} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{5}} &= \frac{1}{12}\left( 436\zeta(9) -279\zeta(3)\zeta(6) -258\zeta(4)\zeta(5) +84\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+40\zeta(3)^3\right) \tag{464} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{4}(k+1)} &= \frac{-1}{144}\left( 4320\zeta(5) +864\zeta(2)\zeta(3) -5874\zeta(6) -432\zeta(3)^2 \right. \nonumber \\ &\left.+3330\zeta(7) +720\zeta(2)\zeta(5) -3096\zeta(3)\zeta(4) +14833\zeta(8) +4032\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-16704\zeta(3)\zeta(5) -3744 M(2,6)\right) \tag{465} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+1)^{2}} &= \frac{-1}{8}\left( -720\zeta(5) -144\zeta(2)\zeta(3) +939\zeta(6) +72\zeta(3)^2 -185\zeta(7) \right. \nonumber \\ &\left.-40\zeta(2)\zeta(5) +172\zeta(3)\zeta(4)\right) \tag{466} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)^{3}} &= \frac{-1}{8}\left( 720\zeta(5) +144\zeta(2)\zeta(3) -899\zeta(6) -72\zeta(3)^2 +109\zeta(7) \right. \nonumber \\ &\left.+40\zeta(2)\zeta(5) -148\zeta(3)\zeta(4)\right) \tag{467} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{4}} &= \frac{1}{144}\left( 4320\zeta(5) +864\zeta(2)\zeta(3) -5154\zeta(6) -432\zeta(3)^2 \right. \nonumber \\ &\left.+1962\zeta(7) +720\zeta(2)\zeta(5) -2664\zeta(3)\zeta(4) +12415\zeta(8) +3312\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-13824\zeta(3)\zeta(5) -3024 M(2,6)\right) \tag{468} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{5}} &= \frac{-1}{12}\left( 174\zeta(9) -99\zeta(3)\zeta(6) -222\zeta(4)\zeta(5) +84\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+32\zeta(3)^3\right) \tag{469} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{4}(k+2)} &= \frac{1}{576}\left( -36 -108\zeta(2) -396\zeta(3) -666\zeta(4) -1080\zeta(5) \right. \nonumber \\ &\left.-216\zeta(2)\zeta(3) +2937\zeta(6) +216\zeta(3)^2 -3330\zeta(7) -720\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+3096\zeta(3)\zeta(4) -29666\zeta(8) -8064\zeta(2)\zeta(3)^2 +33408\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+7488 M(2,6)\right) \tag{470} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+1)(k+2)} &= \frac{-1}{32}\left( 4 +12\zeta(2) +44\zeta(3) +74\zeta(4) -840\zeta(5) \right. \nonumber \\ &\left.-168\zeta(2)\zeta(3) +979\zeta(6) +72\zeta(3)^2 -370\zeta(7) -80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+344\zeta(3)\zeta(4)\right) \tag{471} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)^{2}(k+2)} &= \frac{-1}{16}\left( 4 +12\zeta(2) +44\zeta(3) +74\zeta(4) +600\zeta(5) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3) -899\zeta(6) -72\zeta(3)^2\right) \tag{472} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{3}(k+2)} &= \frac{1}{8}\left( -4 -12\zeta(2) -44\zeta(3) -74\zeta(4) +120\zeta(5) +24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+109\zeta(7) +40\zeta(2)\zeta(5) -148\zeta(3)\zeta(4)\right) \tag{473} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{4}(k+2)} &= \frac{1}{144}\left( -144 -432\zeta(2) -1584\zeta(3) -2664\zeta(4) +5154\zeta(6) \right. \nonumber \\ &\left.+432\zeta(3)^2 +1962\zeta(7) +720\zeta(2)\zeta(5) -2664\zeta(3)\zeta(4) -12415\zeta(8) \right. \nonumber \\ &\left.-3312\zeta(2)\zeta(3)^2 +13824\zeta(3)\zeta(5) +3024 M(2,6)\right) \tag{474} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+2)^{2}} &= \frac{1}{64}\left( 52 +100\zeta(2) +276\zeta(3) +238\zeta(4) +120\zeta(5) \right. \nonumber \\ &\left.+40\zeta(2)\zeta(3) -939\zeta(6) -72\zeta(3)^2 +370\zeta(7) +80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-344\zeta(3)\zeta(4)\right) \tag{475} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)(k+2)^{2}} &= \frac{-1}{4}\left( -7 -14\zeta(2) -40\zeta(3) -39\zeta(4) +90\zeta(5) \right. \nonumber \\ &\left.+16\zeta(2)\zeta(3) -5\zeta(6)\right) \tag{476} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{2}(k+2)^{2}} &= \frac{-1}{16}\left( -60 -124\zeta(2) -364\zeta(3) -386\zeta(4) +120\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3) +859\zeta(6) +72\zeta(3)^2\right) \tag{477} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{3}(k+2)^{2}} &= \frac{1}{8}\left( 64 +136\zeta(2) +408\zeta(3) +460\zeta(4) -240\zeta(5) \right. \nonumber \\ &\left.-32\zeta(2)\zeta(3) -859\zeta(6) -72\zeta(3)^2 -109\zeta(7) -40\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+148\zeta(3)\zeta(4)\right) \tag{478} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+2)^{3}} &= \frac{-1}{64}\left( 332 +388\zeta(2) +740\zeta(3) +6\zeta(4) -456\zeta(5) \right. \nonumber \\ &\left.-152\zeta(2)\zeta(3) -767\zeta(6) -200\zeta(3)^2 +218\zeta(7) +80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-296\zeta(3)\zeta(4)\right) \tag{479} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)(k+2)^{3}} &= \frac{-1}{32}\left( 388 +500\zeta(2) +1060\zeta(3) +318\zeta(4) -1176\zeta(5) \right. \nonumber \\ &\left.-280\zeta(2)\zeta(3) -727\zeta(6) -200\zeta(3)^2 +218\zeta(7) +80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-296\zeta(3)\zeta(4)\right) \tag{480} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{2}(k+2)^{3}} &= \frac{1}{8}\left( -224 -312\zeta(2) -712\zeta(3) -352\zeta(4) +648\zeta(5) \right. \nonumber \\ &\left.+144\zeta(2)\zeta(3) +793\zeta(6) +136\zeta(3)^2 -109\zeta(7) -40\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+148\zeta(3)\zeta(4)\right) \tag{481} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+2)^{4}} &= \frac{1}{576}\left( 12636 +8460\zeta(2) +9180\zeta(3) -11502\zeta(4) -9864\zeta(5) \right. \nonumber \\ &\left.-4968\zeta(2)\zeta(3) -1677\zeta(6) -1944\zeta(3)^2 -6606\zeta(7) -1584\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+6840\zeta(3)\zeta(4) +24830\zeta(8) +6624\zeta(2)\zeta(3)^2 -27648\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-6048 M(2,6)\right) \tag{482} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)(k+2)^{4}} &= \frac{-1}{144}\left( -8064 -6480\zeta(2) -9360\zeta(3) +4320\zeta(4) +10224\zeta(5) \right. \nonumber \\ &\left.+3744\zeta(2)\zeta(3) +4110\zeta(6) +1872\zeta(3)^2 +2322\zeta(7) +432\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-2088\zeta(3)\zeta(4) -12415\zeta(8) -3312\zeta(2)\zeta(3)^2 +13824\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+3024 M(2,6)\right) \tag{483} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+2)^{5}} &= \frac{-1}{24}\left( 1680 +600\zeta(2) +120\zeta(3) -1380\zeta(4) -672\zeta(5) \right. \nonumber \\ &\left.-240\zeta(2)\zeta(3) -318\zeta(6) +144\zeta(3)^2 -570\zeta(7) -336\zeta(2)\zeta(5) +864\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+43\zeta(8) +120\zeta(2)\zeta(3)^2 -288\zeta(3)\zeta(5) +24 M(2,6) +348\zeta(9) -198\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-444\zeta(4)\zeta(5) +168\zeta(2)\zeta(7) +64\zeta(3)^3\right) \tag{484} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{4}} &= \frac{1}{72}\left( 9442\zeta(9) -14685\zeta(3)\zeta(6) +4752\zeta(4)\zeta(5) +2385\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-360\zeta(3)^3\right) \tag{485} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{3}(k+1)} &= \frac{1}{288}\left( 51408\zeta(6) +6480\zeta(3)^2 -36918\zeta(7) -8208\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-9504\zeta(3)\zeta(4) -67811\zeta(8) -19080\zeta(2)\zeta(3)^2 +78768\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+16920 M(2,6)\right) \tag{486} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+1)^{2}} &= \frac{1}{8}\left( -2856\zeta(6) -360\zeta(3)^2 +1953\zeta(7) +456\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+528\zeta(3)\zeta(4)\right) \tag{487} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)^{3}} &= \frac{1}{288}\left( 51408\zeta(6) +6480\zeta(3)^2 -33390\zeta(7) -8208\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-9504\zeta(3)\zeta(4) -65621\zeta(8) -17640\zeta(2)\zeta(3)^2 +72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+15480 M(2,6)\right) \tag{488} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{4}} &= \frac{1}{72}\left( 7120\zeta(9) -12885\zeta(3)\zeta(6) +4752\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+2385\zeta(2)\zeta(7) -360\zeta(3)^3\right) \tag{489} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{3}(k+2)} &= \frac{1}{576}\left( 72 +288\zeta(2) +1512\zeta(3) +4518\zeta(4) +5112\zeta(5) \right. \nonumber \\ &\left.+1080\zeta(2)\zeta(3) +12852\zeta(6) +1620\zeta(3)^2 -18459\zeta(7) -4104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-4752\zeta(3)\zeta(4) -67811\zeta(8) -19080\zeta(2)\zeta(3)^2 +78768\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+16920 M(2,6)\right) \tag{490} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+1)(k+2)} &= \frac{-1}{32}\left( -8 -32\zeta(2) -168\zeta(3) -502\zeta(4) -568\zeta(5) \right. \nonumber \\ &\left.-120\zeta(2)\zeta(3) +4284\zeta(6) +540\zeta(3)^2 -2051\zeta(7) -456\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-528\zeta(3)\zeta(4)\right) \tag{491} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)^{2}(k+2)} &= \frac{-1}{16}\left( -8 -32\zeta(2) -168\zeta(3) -502\zeta(4) -568\zeta(5) \right. \nonumber \\ &\left.-120\zeta(2)\zeta(3) -1428\zeta(6) -180\zeta(3)^2 +1855\zeta(7) +456\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+528\zeta(3)\zeta(4)\right) \tag{492} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{3}(k+2)} &= \frac{1}{288}\left( 288 +1152\zeta(2) +6048\zeta(3) +18072\zeta(4) +20448\zeta(5) \right. \nonumber \\ &\left.+4320\zeta(2)\zeta(3) -33390\zeta(7) -8208\zeta(2)\zeta(5) -9504\zeta(3)\zeta(4) +65621\zeta(8) \right. \nonumber \\ &\left.+17640\zeta(2)\zeta(3)^2 -72432\zeta(3)\zeta(5) -15480 M(2,6)\right) \tag{493} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+2)^{2}} &= \frac{-1}{96}\left( 168 +456\zeta(2) +1896\zeta(3) +3858\zeta(4) +1608\zeta(5) \right. \nonumber \\ &\left.+480\zeta(2)\zeta(3) -11\zeta(6) +180\zeta(3)^2 -5859\zeta(7) -1368\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1584\zeta(3)\zeta(4)\right) \tag{494} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)(k+2)^{2}} &= \frac{-1}{96}\left( 360 +1008\zeta(2) +4296\zeta(3) +9222\zeta(4) +4920\zeta(5) \right. \nonumber \\ &\left.+1320\zeta(2)\zeta(3) -12874\zeta(6) -1260\zeta(3)^2 -5565\zeta(7) -1368\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1584\zeta(3)\zeta(4)\right) \tag{495} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{2}(k+2)^{2}} &= \frac{-1}{24}\left( 192 +552\zeta(2) +2400\zeta(3) +5364\zeta(4) +3312\zeta(5) \right. \nonumber \\ &\left.+840\zeta(2)\zeta(3) -4295\zeta(6) -360\zeta(3)^2 -5565\zeta(7) -1368\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1584\zeta(3)\zeta(4)\right) \tag{496} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+2)^{3}} &= \frac{1}{576}\left( 6984 +12528\zeta(2) +40104\zeta(3) +48726\zeta(4) -13896\zeta(5) \right. \nonumber \\ &\left.-2520\zeta(2)\zeta(3) -58518\zeta(6) -10620\zeta(3)^2 +2925\zeta(7) +3096\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-31392\zeta(3)\zeta(4) -65621\zeta(8) -17640\zeta(2)\zeta(3)^2 +72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+15480 M(2,6)\right) \tag{497} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)(k+2)^{3}} &= \frac{-1}{288}\left( -8064 -15552\zeta(2) -52992\zeta(3) -76392\zeta(4) -864\zeta(5) \right. \nonumber \\ &\left.-1440\zeta(2)\zeta(3) +97140\zeta(6) +14400\zeta(3)^2 +13770\zeta(7) +1008\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+36144\zeta(3)\zeta(4) +65621\zeta(8) +17640\zeta(2)\zeta(3)^2 -72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-15480 M(2,6)\right) \tag{498} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+2)^{4}} &= \frac{1}{144}\left( -8064 -9360\zeta(2) -22320\zeta(3) -11520\zeta(4) +17136\zeta(5) \right. \nonumber \\ &\left.+5760\zeta(2)\zeta(3) +22050\zeta(6) +6480\zeta(3)^2 +900\zeta(7) -720\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+1440\zeta(3)\zeta(4) -62075\zeta(8) -16560\zeta(2)\zeta(3)^2 +69120\zeta(3)\zeta(5) +15120 M(2,6) \right. \nonumber \\ &\left.+14240\zeta(9) -25770\zeta(3)\zeta(6) +9504\zeta(4)\zeta(5) +4770\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-720\zeta(3)^3\right) \tag{499} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{3}} &= \frac{-1}{24}\left( -7474\zeta(9) +13122\zeta(3)\zeta(6) -6048\zeta(4)\zeta(5) -1953\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+544\zeta(3)^3\right) \tag{500} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{2}(k+1)} &= \frac{-1}{8}\left( 5152\zeta(7) +1160\zeta(2)\zeta(5) +2376\zeta(3)\zeta(4) -5843\zeta(8) \right. \nonumber \\ &\left.+328\zeta(2)\zeta(3)^2 -3896\zeta(3)\zeta(5) -456 M(2,6)\right) \tag{501} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+1)^{2}} &= \frac{1}{24}\left( 15456\zeta(7) +3480\zeta(2)\zeta(5) +7128\zeta(3)\zeta(4) -17027\zeta(8) \right. \nonumber \\ &\left.+924\zeta(2)\zeta(3)^2 -11328\zeta(3)\zeta(5) -1308 M(2,6)\right) \tag{502} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)^{3}} &= \frac{-1}{24}\left( 6146\zeta(9) -12582\zeta(3)\zeta(6) +5832\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+1953\zeta(2)\zeta(7) -536\zeta(3)^3\right) \tag{503} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{2}(k+2)} &= \frac{1}{16}\left( -4 -20\zeta(2) -136\zeta(3) -571\zeta(4) -1142\zeta(5) -244\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-2097\zeta(6) -268\zeta(3)^2 -2576\zeta(7) -580\zeta(2)\zeta(5) -1188\zeta(3)\zeta(4) +5843\zeta(8) \right. \nonumber \\ &\left.-328\zeta(2)\zeta(3)^2 +3896\zeta(3)\zeta(5) +456 M(2,6)\right) \tag{504} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+1)(k+2)} &= \frac{-1}{8}\left( 4 +20\zeta(2) +136\zeta(3) +571\zeta(4) +1142\zeta(5) +244\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+2097\zeta(6) +268\zeta(3)^2 -2576\zeta(7) -580\zeta(2)\zeta(5) -1188\zeta(3)\zeta(4)\right) \tag{505} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)^{2}(k+2)} &= \frac{-1}{24}\left( 24 +120\zeta(2) +816\zeta(3) +3426\zeta(4) +6852\zeta(5) \right. \nonumber \\ &\left.+1464\zeta(2)\zeta(3) +12582\zeta(6) +1608\zeta(3)^2 -17027\zeta(8) +924\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-11328\zeta(3)\zeta(5) -1308 M(2,6)\right) \tag{506} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+2)^{2}} &= \frac{-1}{48}\left( -180 -636\zeta(2) -3528\zeta(3) -11001\zeta(4) -13530\zeta(5) \right. \nonumber \\ &\left.-3180\zeta(2)\zeta(3) -5988\zeta(6) -1332\zeta(3)^2 +8967\zeta(7) +2364\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+1188\zeta(3)\zeta(4) +17027\zeta(8) -924\zeta(2)\zeta(3)^2 +11328\zeta(3)\zeta(5) +1308 M(2,6)\right) \tag{507} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)(k+2)^{2}} &= \frac{1}{24}\left( 192 +696\zeta(2) +3936\zeta(3) +12714\zeta(4) +16956\zeta(5) \right. \nonumber \\ &\left.+3912\zeta(2)\zeta(3) +12279\zeta(6) +2136\zeta(3)^2 -16695\zeta(7) -4104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-4752\zeta(3)\zeta(4) -17027\zeta(8) +924\zeta(2)\zeta(3)^2 -11328\zeta(3)\zeta(5) -1308 M(2,6)\right) \tag{508} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+2)^{3}} &= \frac{1}{48}\left( -1344 -3312\zeta(2) -14832\zeta(3) -33120\zeta(4) -20592\zeta(5) \right. \nonumber \\ &\left.-5184\zeta(2)\zeta(3) +24396\zeta(6) +3024\zeta(3)^2 +23580\zeta(7) +4608\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+22824\zeta(3)\zeta(4) +65621\zeta(8) +17640\zeta(2)\zeta(3)^2 -72432\zeta(3)\zeta(5) -15480 M(2,6) \right. \nonumber \\ &\left.-12292\zeta(9) +25164\zeta(3)\zeta(6) -11664\zeta(4)\zeta(5) -3906\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+1072\zeta(3)^3\right) \tag{509} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k^{2}} &= \frac{-1}{72}\left( -276341\zeta(9) -88665\zeta(3)\zeta(6) -143163\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-59166\zeta(2)\zeta(7) -4032\zeta(3)^3\right) \tag{510} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k(k+1)} &= \frac{-1}{18}\left( -119774\zeta(8) -3024\zeta(2)\zeta(3)^2 -27405\zeta(3)\zeta(5)\right) \tag{511} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{(k+1)^{2}} &= \frac{-1}{72}\left( -269402\zeta(9) -88665\zeta(3)\zeta(6) -141273\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-59166\zeta(2)\zeta(7) -4032\zeta(3)^3\right) \tag{512} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{7}}{k(k+2)} &= \frac{1}{288}\left( 144 +864\zeta(2) +7200\zeta(3) +38664\zeta(4) +108504\zeta(5) \right. \nonumber \\ &\left.+23184\zeta(2)\zeta(3) +352887\zeta(6) +45864\zeta(3)^2 +319554\zeta(7) +73080\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+148932\zeta(3)\zeta(4) +958192\zeta(8) +24192\zeta(2)\zeta(3)^2 +219240\zeta(3)\zeta(5)\right) \tag{513} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{(k+1)(k+2)} &= \frac{-1}{48}\left( -48 -288\zeta(2) -2400\zeta(3) -12888\zeta(4) -36168\zeta(5) \right. \nonumber \\ &\left.-7728\zeta(2)\zeta(3) -117629\zeta(6) -15288\zeta(3)^2 -106518\zeta(7) -24360\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-49644\zeta(3)\zeta(4)\right) \tag{514} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{(k+2)^{2}} &= \frac{-1}{144}\left( 1152 +5040\zeta(2) +34992\zeta(3) +146340\zeta(4) +287712\zeta(5) \right. \nonumber \\ &\left.+64512\zeta(2)\zeta(3) +525384\zeta(6) +76608\zeta(3)^2 -31041\zeta(7) -13104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+49140\zeta(3)\zeta(4) -715134\zeta(8) +38808\zeta(2)\zeta(3)^2 -475776\zeta(3)\zeta(5) -54936 M(2,6) \right. \nonumber \\ &\left.-538804\zeta(9) -177330\zeta(3)\zeta(6) -282546\zeta(4)\zeta(5) -118332\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-8064\zeta(3)^3\right) \tag{515} \end{align}\]

Formulas for order \(r = m + n + p + q = 10\): \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{9}} &= \frac{1}{4}\left( 11\zeta(10) -4\zeta(3)\zeta(7) -2\zeta(5)^2\right) \tag{516} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{8}(k+1)} &= \frac{1}{4}\left( -4\zeta(2) +8\zeta(3) -5\zeta(4) +12\zeta(5) -4\zeta(2)\zeta(3) -7\zeta(6) \right. \nonumber \\ &\left.+2\zeta(3)^2 +16\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4) -9\zeta(8) +4\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+20\zeta(9) -4\zeta(3)\zeta(6) -4\zeta(4)\zeta(5) -4\zeta(2)\zeta(7)\right) \tag{517} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{7}(k+1)^{2}} &= \frac{1}{4}\left( 28\zeta(2) -52\zeta(3) +25\zeta(4) -48\zeta(5) +16\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+21\zeta(6) -6\zeta(3)^2 -32\zeta(7) +8\zeta(2)\zeta(5) +8\zeta(3)\zeta(4) +9\zeta(8) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(5)\right) \tag{518} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+1)^{3}} &= \frac{-1}{4}\left( 84\zeta(2) -144\zeta(3) +49\zeta(4) -72\zeta(5) +24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+21\zeta(6) -6\zeta(3)^2 -16\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4)\right) \tag{519} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)^{4}} &= \frac{-1}{4}\left( -140\zeta(2) +220\zeta(3) -45\zeta(4) +56\zeta(5) -20\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-7\zeta(6) +2\zeta(3)^2\right) \tag{520} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{5}} &= \frac{-1}{4}\left( 140\zeta(2) -200\zeta(3) +15\zeta(4) -44\zeta(5) +20\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-3\zeta(6) +2\zeta(3)^2\right) \tag{521} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{6}} &= \frac{-1}{4}\left( -84\zeta(2) +108\zeta(3) +5\zeta(4) +48\zeta(5) -24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+9\zeta(6) -6\zeta(3)^2 +12\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4)\right) \tag{522} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{7}} &= \frac{1}{4}\left( -28\zeta(2) +32\zeta(3) +5\zeta(4) +32\zeta(5) -16\zeta(2)\zeta(3) +9\zeta(6) \right. \nonumber \\ &\left.-6\zeta(3)^2 +24\zeta(7) -8\zeta(2)\zeta(5) -8\zeta(3)\zeta(4) +5\zeta(8) -4\zeta(3)\zeta(5)\right) \tag{523} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{8}} &= \frac{-1}{4}\left( -4\zeta(2) +4\zeta(3) +\zeta(4) +8\zeta(5) -4\zeta(2)\zeta(3) +3\zeta(6) \right. \nonumber \\ &\left.-2\zeta(3)^2 +12\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4) +5\zeta(8) -4\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+16\zeta(9) -4\zeta(3)\zeta(6) -4\zeta(4)\zeta(5) -4\zeta(2)\zeta(7)\right) \tag{524} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{9}} &= \frac{-1}{4}\left( -7\zeta(10) +4\zeta(3)\zeta(7) +2\zeta(5)^2\right) \tag{525} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{8}(k+2)} &= \frac{-1}{256}\left( 1 +\zeta(2) -4\zeta(3) +5\zeta(4) -24\zeta(5) +8\zeta(2)\zeta(3) +28\zeta(6) \right. \nonumber \\ &\left.-8\zeta(3)^2 -128\zeta(7) +32\zeta(2)\zeta(5) +32\zeta(3)\zeta(4) +144\zeta(8) -64\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-640\zeta(9) +128\zeta(3)\zeta(6) +128\zeta(4)\zeta(5) +128\zeta(2)\zeta(7)\right) \tag{526} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{7}(k+1)(k+2)} &= \frac{1}{128}\left( -1 +127\zeta(2) -252\zeta(3) +155\zeta(4) -360\zeta(5) +120\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+196\zeta(6) -56\zeta(3)^2 -384\zeta(7) +96\zeta(2)\zeta(5) +96\zeta(3)\zeta(4) +144\zeta(8) \right. \nonumber \\ &\left.-64\zeta(3)\zeta(5)\right) \tag{527} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+1)^{2}(k+2)} &= \frac{1}{64}\left( -1 -321\zeta(2) +580\zeta(3) -245\zeta(4) +408\zeta(5) \right. \nonumber \\ &\left.-136\zeta(2)\zeta(3) -140\zeta(6) +40\zeta(3)^2 +128\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-32\zeta(3)\zeta(4)\right) \tag{528} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)^{3}(k+2)} &= \frac{-1}{32}\left( 1 -351\zeta(2) +572\zeta(3) -147\zeta(4) +168\zeta(5) \right. \nonumber \\ &\left.-56\zeta(2)\zeta(3) -28\zeta(6) +8\zeta(3)^2\right) \tag{529} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{4}(k+2)} &= \frac{1}{16}\left( -1 -209\zeta(2) +308\zeta(3) -33\zeta(4) +56\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(3)\right) \tag{530} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{5}(k+2)} &= \frac{-1}{8}\left( 1 -71\zeta(2) +92\zeta(3) +3\zeta(4) +32\zeta(5) -16\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+6\zeta(6) -4\zeta(3)^2\right) \tag{531} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{6}(k+2)} &= \frac{-1}{4}\left( 1 +13\zeta(2) -16\zeta(3) -2\zeta(4) -16\zeta(5) +8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-3\zeta(6) +2\zeta(3)^2 -12\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4)\right) \tag{532} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{7}(k+2)} &= \frac{-1}{4}\left( 2 -2\zeta(2) +\zeta(4) +3\zeta(6) -2\zeta(3)^2 +5\zeta(8) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(5)\right) \tag{533} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{8}(k+2)} &= \frac{1}{4}\left( -4 +4\zeta(3) -\zeta(4) +8\zeta(5) -4\zeta(2)\zeta(3) -3\zeta(6) +2\zeta(3)^2 \right. \nonumber \\ &\left.+12\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4) -5\zeta(8) +4\zeta(3)\zeta(5) +16\zeta(9) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(6) -4\zeta(4)\zeta(5) -4\zeta(2)\zeta(7)\right) \tag{534} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{7}(k+2)^{2}} &= \frac{1}{256}\left( 11 +5\zeta(2) -26\zeta(3) +25\zeta(4) -96\zeta(5) +32\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+84\zeta(6) -24\zeta(3)^2 -256\zeta(7) +64\zeta(2)\zeta(5) +64\zeta(3)\zeta(4) +144\zeta(8) \right. \nonumber \\ &\left.-64\zeta(3)\zeta(5)\right) \tag{535} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+1)(k+2)^{2}} &= \frac{-1}{64}\left( -6 +61\zeta(2) -113\zeta(3) +65\zeta(4) -132\zeta(5) \right. \nonumber \\ &\left.+44\zeta(2)\zeta(3) +56\zeta(6) -16\zeta(3)^2 -64\zeta(7) +16\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+16\zeta(3)\zeta(4)\right) \tag{536} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)^{2}(k+2)^{2}} &= \frac{1}{64}\left( 13 +199\zeta(2) -354\zeta(3) +115\zeta(4) -144\zeta(5) \right. \nonumber \\ &\left.+48\zeta(2)\zeta(3) +28\zeta(6) -8\zeta(3)^2\right) \tag{537} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{3}(k+2)^{2}} &= \frac{-1}{16}\left( -7 +76\zeta(2) -109\zeta(3) +16\zeta(4) -12\zeta(5) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(3)\right) \tag{538} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{4}(k+2)^{2}} &= \frac{-1}{16}\left( -15 -57\zeta(2) +90\zeta(3) -\zeta(4) +32\zeta(5) \right. \nonumber \\ &\left.-16\zeta(2)\zeta(3)\right) \tag{539} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{5}(k+2)^{2}} &= \frac{1}{4}\left( 8 -7\zeta(2) +\zeta(3) +2\zeta(4) +3\zeta(6) -2\zeta(3)^2\right) \tag{540} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{6}(k+2)^{2}} &= \frac{1}{4}\left( 17 -\zeta(2) -14\zeta(3) +2\zeta(4) -16\zeta(5) +8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+3\zeta(6) -2\zeta(3)^2 -12\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4)\right) \tag{541} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{7}(k+2)^{2}} &= \frac{1}{4}\left( 36 -4\zeta(2) -28\zeta(3) +5\zeta(4) -32\zeta(5) +16\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+9\zeta(6) -6\zeta(3)^2 -24\zeta(7) +8\zeta(2)\zeta(5) +8\zeta(3)\zeta(4) +5\zeta(8) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(5)\right) \tag{542} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+2)^{3}} &= \frac{1}{256}\left( -57 -5\zeta(2) +76\zeta(3) -49\zeta(4) +144\zeta(5) -48\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-84\zeta(6) +24\zeta(3)^2 +128\zeta(7) -32\zeta(2)\zeta(5) -32\zeta(3)\zeta(4)\right) \tag{543} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)(k+2)^{3}} &= \frac{1}{128}\left( -69 +117\zeta(2) -150\zeta(3) +81\zeta(4) -120\zeta(5) \right. \nonumber \\ &\left.+40\zeta(2)\zeta(3) +28\zeta(6) -8\zeta(3)^2\right) \tag{544} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{2}(k+2)^{3}} &= \frac{1}{32}\left( -41 -41\zeta(2) +102\zeta(3) -17\zeta(4) +12\zeta(5) \right. \nonumber \\ &\left.-4\zeta(2)\zeta(3)\right) \tag{545} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{3}(k+2)^{3}} &= \frac{-1}{16}\left( 48 -35\zeta(2) +7\zeta(3) +\zeta(4)\right) \tag{546} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{4}(k+2)^{3}} &= \frac{1}{16}\left( -111 +13\zeta(2) +76\zeta(3) -3\zeta(4) +32\zeta(5) \right. \nonumber \\ &\left.-16\zeta(2)\zeta(3)\right) \tag{547} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{5}(k+2)^{3}} &= \frac{-1}{8}\left( 127 -27\zeta(2) -74\zeta(3) +7\zeta(4) -32\zeta(5) +16\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+6\zeta(6) -4\zeta(3)^2\right) \tag{548} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{6}(k+2)^{3}} &= \frac{-1}{4}\left( 144 -28\zeta(2) -88\zeta(3) +9\zeta(4) -48\zeta(5) +24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+9\zeta(6) -6\zeta(3)^2 -12\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4)\right) \tag{549} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+2)^{4}} &= \frac{1}{256}\left( 187 -23\zeta(2) -138\zeta(3) +37\zeta(4) -112\zeta(5) +40\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+28\zeta(6) -8\zeta(3)^2\right) \tag{550} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)(k+2)^{4}} &= \frac{-1}{32}\left( -64 +35\zeta(2) -3\zeta(3) +11\zeta(4) -2\zeta(5)\right) \tag{551} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{2}(k+2)^{4}} &= \frac{1}{32}\left( 169 -29\zeta(2) -96\zeta(3) -5\zeta(4) -8\zeta(5) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(3)\right) \tag{552} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{3}(k+2)^{4}} &= \frac{-1}{16}\left( -217 +64\zeta(2) +89\zeta(3) +4\zeta(4) +8\zeta(5) \right. \nonumber \\ &\left.-4\zeta(2)\zeta(3)\right) \tag{553} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{4}(k+2)^{4}} &= \frac{1}{16}\left( 545 -141\zeta(2) -254\zeta(3) -5\zeta(4) -48\zeta(5) \right. \nonumber \\ &\left.+24\zeta(2)\zeta(3)\right) \tag{554} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{5}(k+2)^{4}} &= \frac{-1}{4}\left( -336 +84\zeta(2) +164\zeta(3) -\zeta(4) +40\zeta(5) -20\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-3\zeta(6) +2\zeta(3)^2\right) \tag{555} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+2)^{5}} &= \frac{1}{256}\left( -443 +93\zeta(2) +188\zeta(3) +33\zeta(4) +104\zeta(5) -40\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+12\zeta(6) -8\zeta(3)^2\right) \tag{556} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)(k+2)^{5}} &= \frac{-1}{128}\left( 699 -233\zeta(2) -176\zeta(3) -77\zeta(4) -96\zeta(5) \right. \nonumber \\ &\left.+40\zeta(2)\zeta(3) -12\zeta(6) +8\zeta(3)^2\right) \tag{557} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{2}(k+2)^{5}} &= \frac{1}{64}\left( -1037 +291\zeta(2) +368\zeta(3) +87\zeta(4) +112\zeta(5) \right. \nonumber \\ &\left.-48\zeta(2)\zeta(3) +12\zeta(6) -8\zeta(3)^2\right) \tag{558} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{3}(k+2)^{5}} &= \frac{1}{32}\left( -1471 +419\zeta(2) +546\zeta(3) +95\zeta(4) +128\zeta(5) \right. \nonumber \\ &\left.-56\zeta(2)\zeta(3) +12\zeta(6) -8\zeta(3)^2\right) \tag{559} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{4}(k+2)^{5}} &= \frac{-1}{4}\left( 504 -140\zeta(2) -200\zeta(3) -25\zeta(4) -44\zeta(5) \right. \nonumber \\ &\left.+20\zeta(2)\zeta(3) -3\zeta(6) +2\zeta(3)^2\right) \tag{560} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+2)^{6}} &= \frac{1}{256}\left( 825 -177\zeta(2) -222\zeta(3) -133\zeta(4) -176\zeta(5) +48\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-68\zeta(6) +24\zeta(3)^2 -96\zeta(7) +32\zeta(2)\zeta(5) +32\zeta(3)\zeta(4)\right) \tag{561} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)(k+2)^{6}} &= \frac{-1}{64}\left( -762 +205\zeta(2) +199\zeta(3) +105\zeta(4) +136\zeta(5) \right. \nonumber \\ &\left.-44\zeta(2)\zeta(3) +40\zeta(6) -16\zeta(3)^2 +48\zeta(7) -16\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-16\zeta(3)\zeta(4)\right) \tag{562} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{2}(k+2)^{6}} &= \frac{-1}{64}\left( -2561 +701\zeta(2) +766\zeta(3) +297\zeta(4) +384\zeta(5) \right. \nonumber \\ &\left.-136\zeta(2)\zeta(3) +92\zeta(6) -40\zeta(3)^2 +96\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-32\zeta(3)\zeta(4)\right) \tag{563} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{3}(k+2)^{6}} &= \frac{-1}{4}\left( -504 +140\zeta(2) +164\zeta(3) +49\zeta(4) +64\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(3) +13\zeta(6) -6\zeta(3)^2 +12\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4)\right) \tag{564} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+2)^{7}} &= \frac{1}{256}\left( -1291 +233\zeta(2) +244\zeta(3) +213\zeta(4) +240\zeta(5) \right. \nonumber \\ &\left.-32\zeta(2)\zeta(3) +164\zeta(6) -24\zeta(3)^2 +256\zeta(7) -64\zeta(2)\zeta(5) -64\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+80\zeta(8) -64\zeta(3)\zeta(5)\right) \tag{565} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)(k+2)^{7}} &= \frac{-1}{128}\left( 2815 -643\zeta(2) -642\zeta(3) -423\zeta(4) -512\zeta(5) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3) -244\zeta(6) +56\zeta(3)^2 -352\zeta(7) +96\zeta(2)\zeta(5) +96\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-80\zeta(8) +64\zeta(3)\zeta(5)\right) \tag{566} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{2}(k+2)^{7}} &= \frac{1}{4}\left( -336 +84\zeta(2) +88\zeta(3) +45\zeta(4) +56\zeta(5) -16\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+21\zeta(6) -6\zeta(3)^2 +28\zeta(7) -8\zeta(2)\zeta(5) -8\zeta(3)\zeta(4) +5\zeta(8) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(5)\right) \tag{567} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k(k+2)^{8}} &= \frac{-1}{256}\left( -1793 +253\zeta(2) +254\zeta(3) +249\zeta(4) +256\zeta(5) -8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+236\zeta(6) -8\zeta(3)^2 +288\zeta(7) -32\zeta(2)\zeta(5) -32\zeta(3)\zeta(4) +208\zeta(8) \right. \nonumber \\ &\left.-64\zeta(3)\zeta(5) +512\zeta(9) -128\zeta(3)\zeta(6) -128\zeta(4)\zeta(5) -128\zeta(2)\zeta(7)\right) \tag{568} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)(k+2)^{8}} &= \frac{1}{4}\left( 144 -28\zeta(2) -28\zeta(3) -21\zeta(4) -24\zeta(5) +4\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-15\zeta(6) +2\zeta(3)^2 -20\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4) -9\zeta(8) \right. \nonumber \\ &\left.+4\zeta(3)\zeta(5) -16\zeta(9) +4\zeta(3)\zeta(6) +4\zeta(4)\zeta(5) +4\zeta(2)\zeta(7)\right) \tag{569} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+2)^{9}} &= \frac{-1}{4}\left( 36 -4\zeta(2) -4\zeta(3) -4\zeta(4) -4\zeta(5) -4\zeta(6) -4\zeta(7) -4\zeta(8) \right. \nonumber \\ &\left.-4\zeta(9) -7\zeta(10) +4\zeta(3)\zeta(7) +2\zeta(5)^2\right) \tag{570} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{8}} &= -\left( - M(2,8)\right) \tag{571} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{7}(k+1)} &= \frac{-1}{24}\left( -72\zeta(3) +102\zeta(4) -84\zeta(5) +24\zeta(2)\zeta(3) +97\zeta(6) \right. \nonumber \\ &\left.-48\zeta(3)^2 -144\zeta(7) +24\zeta(2)\zeta(5) +60\zeta(3)\zeta(4) +24 M(2,6) -220\zeta(9) \right. \nonumber \\ &\left.+84\zeta(3)\zeta(6) +60\zeta(4)\zeta(5) +24\zeta(2)\zeta(7) -8\zeta(3)^3\right) \tag{572} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{6}(k+1)^{2}} &= \frac{1}{8}\left( -144\zeta(3) +192\zeta(4) -112\zeta(5) +32\zeta(2)\zeta(3) +97\zeta(6) \right. \nonumber \\ &\left.-48\zeta(3)^2 -96\zeta(7) +16\zeta(2)\zeta(5) +40\zeta(3)\zeta(4) +8 M(2,6)\right) \tag{573} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+1)^{3}} &= \frac{1}{8}\left( 360\zeta(3) -450\zeta(4) +180\zeta(5) -56\zeta(2)\zeta(3) -97\zeta(6) \right. \nonumber \\ &\left.+48\zeta(3)^2 +48\zeta(7) -8\zeta(2)\zeta(5) -20\zeta(3)\zeta(4)\right) \tag{574} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)^{4}} &= \frac{-1}{12}\left( 720\zeta(3) -840\zeta(4) +240\zeta(5) -96\zeta(2)\zeta(3) -67\zeta(6) \right. \nonumber \\ &\left.+36\zeta(3)^2\right) \tag{575} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{5}} &= \frac{-1}{8}\left( -360\zeta(3) +390\zeta(4) -100\zeta(5) +56\zeta(2)\zeta(3) +37\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2 -8\zeta(7) +8\zeta(2)\zeta(5) -4\zeta(3)\zeta(4)\right) \tag{576} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{6}} &= \frac{-1}{8}\left( 144\zeta(3) -144\zeta(4) +48\zeta(5) -32\zeta(2)\zeta(3) -37\zeta(6) \right. \nonumber \\ &\left.+24\zeta(3)^2 +16\zeta(7) -16\zeta(2)\zeta(5) +8\zeta(3)\zeta(4) +28\zeta(8) -16\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-8 M(2,6)\right) \tag{577} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{7}} &= \frac{1}{24}\left( 72\zeta(3) -66\zeta(4) +36\zeta(5) -24\zeta(2)\zeta(3) -37\zeta(6) \right. \nonumber \\ &\left.+24\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) +12\zeta(3)\zeta(4) +84\zeta(8) -48\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-24 M(2,6) -4\zeta(9) +36\zeta(3)\zeta(6) +12\zeta(4)\zeta(5) -24\zeta(2)\zeta(7) -8\zeta(3)^3\right) \tag{578} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{8}} &= \frac{1}{2}\left( -9\zeta(10) +4\zeta(3)\zeta(7) +2\zeta(5)^2 +2 M(2,8)\right) \tag{579} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{7}(k+2)} &= \frac{-1}{768}\left( -6 -6\zeta(2) -18\zeta(3) +51\zeta(4) -84\zeta(5) +24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+194\zeta(6) -96\zeta(3)^2 -576\zeta(7) +96\zeta(2)\zeta(5) +240\zeta(3)\zeta(4) +192 M(2,6) \right. \nonumber \\ &\left.-3520\zeta(9) +1344\zeta(3)\zeta(6) +960\zeta(4)\zeta(5) +384\zeta(2)\zeta(7) -128\zeta(3)^3\right) \tag{580} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{6}(k+1)(k+2)} &= \frac{-1}{384}\left( -6 -6\zeta(2) +1134\zeta(3) -1581\zeta(4) +1260\zeta(5) \right. \nonumber \\ &\left.-360\zeta(2)\zeta(3) -1358\zeta(6) +672\zeta(3)^2 +1728\zeta(7) -288\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-720\zeta(3)\zeta(4) -192 M(2,6)\right) \tag{581} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+1)^{2}(k+2)} &= \frac{-1}{192}\left( -6 -6\zeta(2) -2322\zeta(3) +3027\zeta(4) -1428\zeta(5) \right. \nonumber \\ &\left.+408\zeta(2)\zeta(3) +970\zeta(6) -480\zeta(3)^2 -576\zeta(7) +96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+240\zeta(3)\zeta(4)\right) \tag{582} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)^{3}(k+2)} &= \frac{-1}{96}\left( -6 -6\zeta(2) +1998\zeta(3) -2373\zeta(4) +732\zeta(5) \right. \nonumber \\ &\left.-264\zeta(2)\zeta(3) -194\zeta(6) +96\zeta(3)^2\right) \tag{583} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{4}(k+2)} &= \frac{-1}{48}\left( -6 -6\zeta(2) -882\zeta(3) +987\zeta(4) -228\zeta(5) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3) +74\zeta(6) -48\zeta(3)^2\right) \tag{584} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{5}(k+2)} &= \frac{-1}{24}\left( -6 -6\zeta(2) +198\zeta(3) -183\zeta(4) +72\zeta(5) \right. \nonumber \\ &\left.-48\zeta(2)\zeta(3) -37\zeta(6) +24\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+12\zeta(3)\zeta(4)\right) \tag{585} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{6}(k+2)} &= \frac{-1}{24}\left( -12 -12\zeta(2) -36\zeta(3) +66\zeta(4) +37\zeta(6) -24\zeta(3)^2 \right. \nonumber \\ &\left.-84\zeta(8) +48\zeta(3)\zeta(5) +24 M(2,6)\right) \tag{586} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{7}(k+2)} &= \frac{-1}{24}\left( -24 -24\zeta(2) +66\zeta(4) +36\zeta(5) -24\zeta(2)\zeta(3) +37\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) +12\zeta(3)\zeta(4) -84\zeta(8) +48\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+24 M(2,6) -4\zeta(9) +36\zeta(3)\zeta(6) +12\zeta(4)\zeta(5) -24\zeta(2)\zeta(7) -8\zeta(3)^3\right) \tag{587} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{6}(k+2)^{2}} &= \frac{1}{128}\left( -12 -6\zeta(2) -14\zeta(3) +48\zeta(4) -56\zeta(5) +16\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+97\zeta(6) -48\zeta(3)^2 -192\zeta(7) +32\zeta(2)\zeta(5) +80\zeta(3)\zeta(4) +32 M(2,6)\right) \tag{588} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+1)(k+2)^{2}} &= \frac{1}{384}\left( -78 -42\zeta(2) +1050\zeta(3) -1293\zeta(4) +924\zeta(5) \right. \nonumber \\ &\left.-264\zeta(2)\zeta(3) -776\zeta(6) +384\zeta(3)^2 +576\zeta(7) -96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-240\zeta(3)\zeta(4)\right) \tag{589} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)^{2}(k+2)^{2}} &= \frac{1}{96}\left( -42 -24\zeta(2) -636\zeta(3) +867\zeta(4) -252\zeta(5) \right. \nonumber \\ &\left.+72\zeta(2)\zeta(3) +97\zeta(6) -48\zeta(3)^2\right) \tag{590} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{3}(k+2)^{2}} &= \frac{1}{32}\left( -30 -18\zeta(2) +242\zeta(3) -213\zeta(4) +76\zeta(5) \right. \nonumber \\ &\left.-40\zeta(2)\zeta(3)\right) \tag{591} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{4}(k+2)^{2}} &= \frac{-1}{24}\left( 48 +30\zeta(2) +78\zeta(3) -174\zeta(4) -37\zeta(6) \right. \nonumber \\ &\left.+24\zeta(3)^2\right) \tag{592} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{5}(k+2)^{2}} &= \frac{-1}{24}\left( 102 +66\zeta(2) -42\zeta(3) -165\zeta(4) -72\zeta(5) \right. \nonumber \\ &\left.+48\zeta(2)\zeta(3) -37\zeta(6) +24\zeta(3)^2 -24\zeta(7) +24\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-12\zeta(3)\zeta(4)\right) \tag{593} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{6}(k+2)^{2}} &= \frac{-1}{8}\left( 72 +48\zeta(2) -16\zeta(3) -132\zeta(4) -48\zeta(5) \right. \nonumber \\ &\left.+32\zeta(2)\zeta(3) -37\zeta(6) +24\zeta(3)^2 -16\zeta(7) +16\zeta(2)\zeta(5) -8\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+28\zeta(8) -16\zeta(3)\zeta(5) -8 M(2,6)\right) \tag{594} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+2)^{3}} &= \frac{1}{256}\left( 138 +22\zeta(2) +26\zeta(3) -229\zeta(4) +180\zeta(5) \right. \nonumber \\ &\left.-56\zeta(2)\zeta(3) -194\zeta(6) +96\zeta(3)^2 +192\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-80\zeta(3)\zeta(4)\right) \tag{595} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)(k+2)^{3}} &= \frac{1}{192}\left( 246 +54\zeta(2) -486\zeta(3) +303\zeta(4) -192\zeta(5) \right. \nonumber \\ &\left.+48\zeta(2)\zeta(3) +97\zeta(6) -48\zeta(3)^2\right) \tag{596} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{2}(k+2)^{3}} &= \frac{1}{16}\left( 48 +13\zeta(2) +25\zeta(3) -94\zeta(4) +10\zeta(5) \right. \nonumber \\ &\left.-4\zeta(2)\zeta(3)\right) \tag{597} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{3}(k+2)^{3}} &= \frac{-1}{32}\left( -222 -70\zeta(2) +142\zeta(3) +163\zeta(4) +36\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(3)\right) \tag{598} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{4}(k+2)^{3}} &= \frac{1}{48}\left( 762 +270\zeta(2) -270\zeta(3) -837\zeta(4) -108\zeta(5) \right. \nonumber \\ &\left.+72\zeta(2)\zeta(3) -74\zeta(6) +48\zeta(3)^2\right) \tag{599} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{5}(k+2)^{3}} &= \frac{-1}{8}\left( -288 -112\zeta(2) +104\zeta(3) +334\zeta(4) +60\zeta(5) \right. \nonumber \\ &\left.-40\zeta(2)\zeta(3) +37\zeta(6) -24\zeta(3)^2 +8\zeta(7) -8\zeta(2)\zeta(5) +4\zeta(3)\zeta(4)\right) \tag{600} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+2)^{4}} &= \frac{-1}{192}\left( 384 -18\zeta(2) -90\zeta(3) -240\zeta(4) +72\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(3) -67\zeta(6) +36\zeta(3)^2\right) \tag{601} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)(k+2)^{4}} &= \frac{-1}{192}\left( 1014 +18\zeta(2) -666\zeta(3) -177\zeta(4) -48\zeta(5) \right. \nonumber \\ &\left.-37\zeta(6) +24\zeta(3)^2\right) \tag{602} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{2}(k+2)^{4}} &= \frac{-1}{96}\left( 1302 +96\zeta(2) -516\zeta(3) -741\zeta(4) +12\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(3) -37\zeta(6) +24\zeta(3)^2\right) \tag{603} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{3}(k+2)^{4}} &= \frac{-1}{96}\left( 3270 +402\zeta(2) -1458\zeta(3) -1971\zeta(4) -84\zeta(5) \right. \nonumber \\ &\left.+24\zeta(2)\zeta(3) -74\zeta(6) +48\zeta(3)^2\right) \tag{604} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{4}(k+2)^{4}} &= \frac{1}{12}\left( -1008 -168\zeta(2) +432\zeta(3) +702\zeta(4) +48\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(3) +37\zeta(6) -24\zeta(3)^2\right) \tag{605} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+2)^{5}} &= \frac{1}{256}\left( 1398 -210\zeta(2) -486\zeta(3) -371\zeta(4) -252\zeta(5) \right. \nonumber \\ &\left.+104\zeta(2)\zeta(3) -122\zeta(6) +80\zeta(3)^2 +32\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+16\zeta(3)\zeta(4)\right) \tag{606} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)(k+2)^{5}} &= \frac{-1}{384}\left( -6222 +594\zeta(2) +2790\zeta(3) +1467\zeta(4) +852\zeta(5) \right. \nonumber \\ &\left.-312\zeta(2)\zeta(3) +440\zeta(6) -288\zeta(3)^2 -96\zeta(7) +96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-48\zeta(3)\zeta(4)\right) \tag{607} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{2}(k+2)^{5}} &= \frac{-1}{192}\left( -8826 +402\zeta(2) +3822\zeta(3) +2949\zeta(4) +828\zeta(5) \right. \nonumber \\ &\left.-264\zeta(2)\zeta(3) +514\zeta(6) -336\zeta(3)^2 -96\zeta(7) +96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-48\zeta(3)\zeta(4)\right) \tag{608} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{3}(k+2)^{5}} &= \frac{-1}{8}\left( -1008 +440\zeta(3) +410\zeta(4) +76\zeta(5) -24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+49\zeta(6) -32\zeta(3)^2 -8\zeta(7) +8\zeta(2)\zeta(5) -4\zeta(3)\zeta(4)\right) \tag{609} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+2)^{6}} &= \frac{-1}{128}\left( 1524 -282\zeta(2) -498\zeta(3) -272\zeta(4) -424\zeta(5) \right. \nonumber \\ &\left.+160\zeta(2)\zeta(3) -165\zeta(6) +88\zeta(3)^2 -160\zeta(7) +32\zeta(2)\zeta(5) +80\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+112\zeta(8) -64\zeta(3)\zeta(5) -32 M(2,6)\right) \tag{610} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)(k+2)^{6}} &= \frac{1}{384}\left( -15366 +2286\zeta(2) +5778\zeta(3) +3099\zeta(4) +3396\zeta(5) \right. \nonumber \\ &\left.-1272\zeta(2)\zeta(3) +1430\zeta(6) -816\zeta(3)^2 +864\zeta(7) -96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-528\zeta(3)\zeta(4) -672\zeta(8) +384\zeta(3)\zeta(5) +192 M(2,6)\right) \tag{611} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{2}(k+2)^{6}} &= \frac{-1}{8}\left( 1008 -112\zeta(2) -400\zeta(3) -252\zeta(4) -176\zeta(5) \right. \nonumber \\ &\left.+64\zeta(2)\zeta(3) -81\zeta(6) +48\zeta(3)^2 -32\zeta(7) +24\zeta(3)\zeta(4) +28\zeta(8) \right. \nonumber \\ &\left.-16\zeta(3)\zeta(5) -8 M(2,6)\right) \tag{612} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+2)^{7}} &= \frac{1}{768}\left( 16890 -3090\zeta(2) -4590\zeta(3) -2757\zeta(4) -4476\zeta(5) \right. \nonumber \\ &\left.+1416\zeta(2)\zeta(3) -2042\zeta(6) +720\zeta(3)^2 -3744\zeta(7) +1056\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+1200\zeta(3)\zeta(4) -288\zeta(8) +384\zeta(3)\zeta(5) -192 M(2,6) -64\zeta(9) +576\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+192\zeta(4)\zeta(5) -384\zeta(2)\zeta(7) -128\zeta(3)^3\right) \tag{613} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)(k+2)^{7}} &= \frac{-1}{24}\left( -2016 +336\zeta(2) +648\zeta(3) +366\zeta(4) +492\zeta(5) \right. \nonumber \\ &\left.-168\zeta(2)\zeta(3) +217\zeta(6) -96\zeta(3)^2 +288\zeta(7) -72\zeta(2)\zeta(5) -108\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-24\zeta(8) +24 M(2,6) +4\zeta(9) -36\zeta(3)\zeta(6) -12\zeta(4)\zeta(5) +24\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+8\zeta(3)^3\right) \tag{614} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+2)^{8}} &= \frac{1}{2}\left( -72 +12\zeta(2) +16\zeta(3) +11\zeta(4) +16\zeta(5) -4\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+9\zeta(6) -2\zeta(3)^2 +16\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4) +7\zeta(8) -4\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+16\zeta(9) -4\zeta(3)\zeta(6) -4\zeta(4)\zeta(5) -4\zeta(2)\zeta(7) -9\zeta(10) +4\zeta(3)\zeta(7) \right. \nonumber \\ &\left.+2\zeta(5)^2 +2 M(2,8)\right) \tag{615} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{7}} &= \frac{-1}{160}\left( 1661\zeta(10) -1280\zeta(3)\zeta(7) -80\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+560\zeta(2)\zeta(3)\zeta(5) -720\zeta(5)^2 -520 M(2,8)\right) \tag{616} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{6}(k+1)} &= \frac{1}{96}\left( -960\zeta(4) +960\zeta(5) +96\zeta(2)\zeta(3) -558\zeta(6) +240\zeta(3)^2 \right. \nonumber \\ &\left.+1386\zeta(7) +192\zeta(2)\zeta(5) -1224\zeta(3)\zeta(4) +595\zeta(8) +120\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-576\zeta(3)\zeta(5) -264 M(2,6) +2084\zeta(9) -1164\zeta(3)\zeta(6) -1224\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+288\zeta(2)\zeta(7) +192\zeta(3)^3\right) \tag{617} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{5}(k+1)^{2}} &= \frac{-1}{96}\left( -4800\zeta(4) +4560\zeta(5) +480\zeta(2)\zeta(3) -1674\zeta(6) \right. \nonumber \\ &\left.+720\zeta(3)^2 +2772\zeta(7) +384\zeta(2)\zeta(5) -2448\zeta(3)\zeta(4) +595\zeta(8) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3)^2 -576\zeta(3)\zeta(5) -264 M(2,6)\right) \tag{618} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+1)^{3}} &= \frac{-1}{16}\left( 1600\zeta(4) -1440\zeta(5) -160\zeta(2)\zeta(3) +312\zeta(6) \right. \nonumber \\ &\left.-152\zeta(3)^2 -231\zeta(7) -32\zeta(2)\zeta(5) +204\zeta(3)\zeta(4)\right) \tag{619} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)^{4}} &= \frac{1}{16}\left( 1600\zeta(4) -1360\zeta(5) -160\zeta(2)\zeta(3) +192\zeta(6) \right. \nonumber \\ &\left.-136\zeta(3)^2 -119\zeta(7) -32\zeta(2)\zeta(5) +132\zeta(3)\zeta(4)\right) \tag{620} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{5}} &= \frac{-1}{96}\left( 4800\zeta(4) -3840\zeta(5) -480\zeta(2)\zeta(3) +594\zeta(6) \right. \nonumber \\ &\left.-576\zeta(3)^2 -1428\zeta(7) -384\zeta(2)\zeta(5) +1584\zeta(3)\zeta(4) +43\zeta(8) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3)^2 -288\zeta(3)\zeta(5) +24 M(2,6)\right) \tag{621} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{6}} &= \frac{1}{96}\left( 960\zeta(4) -720\zeta(5) -96\zeta(2)\zeta(3) +198\zeta(6) -192\zeta(3)^2 \right. \nonumber \\ &\left.-714\zeta(7) -192\zeta(2)\zeta(5) +792\zeta(3)\zeta(4) +43\zeta(8) +120\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-288\zeta(3)\zeta(5) +24 M(2,6) -788\zeta(9) +444\zeta(3)\zeta(6) +792\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-288\zeta(2)\zeta(7) -96\zeta(3)^3\right) \tag{622} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{7}} &= \frac{1}{160}\left( -501\zeta(10) +800\zeta(3)\zeta(7) +80\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.-560\zeta(2)\zeta(3)\zeta(5) +480\zeta(5)^2 +40 M(2,8)\right) \tag{623} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{6}(k+2)} &= \frac{1}{768}\left( -12 -24\zeta(2) -48\zeta(3) -120\zeta(4) +240\zeta(5) +24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-279\zeta(6) +120\zeta(3)^2 +1386\zeta(7) +192\zeta(2)\zeta(5) -1224\zeta(3)\zeta(4) +1190\zeta(8) \right. \nonumber \\ &\left.+240\zeta(2)\zeta(3)^2 -1152\zeta(3)\zeta(5) -528 M(2,6) +8336\zeta(9) -4656\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-4896\zeta(4)\zeta(5) +1152\zeta(2)\zeta(7) +768\zeta(3)^3\right) \tag{624} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{5}(k+1)(k+2)} &= \frac{1}{384}\left( -12 -24\zeta(2) -48\zeta(3) +3720\zeta(4) -3600\zeta(5) \right. \nonumber \\ &\left.-360\zeta(2)\zeta(3) +1953\zeta(6) -840\zeta(3)^2 -4158\zeta(7) -576\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+3672\zeta(3)\zeta(4) -1190\zeta(8) -240\zeta(2)\zeta(3)^2 +1152\zeta(3)\zeta(5) +528 M(2,6)\right) \tag{625} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+1)^{2}(k+2)} &= \frac{1}{64}\left( -4 -8\zeta(2) -16\zeta(3) -1960\zeta(4) +1840\zeta(5) \right. \nonumber \\ &\left.+200\zeta(2)\zeta(3) -465\zeta(6) +200\zeta(3)^2 +462\zeta(7) +64\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-408\zeta(3)\zeta(4)\right) \tag{626} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)^{3}(k+2)} &= \frac{1}{32}\left( -4 -8\zeta(2) -16\zeta(3) +1240\zeta(4) -1040\zeta(5) \right. \nonumber \\ &\left.-120\zeta(2)\zeta(3) +159\zeta(6) -104\zeta(3)^2\right) \tag{627} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{4}(k+2)} &= \frac{-1}{16}\left( 4 +8\zeta(2) +16\zeta(3) +360\zeta(4) -320\zeta(5) \right. \nonumber \\ &\left.-40\zeta(2)\zeta(3) +33\zeta(6) -32\zeta(3)^2 -119\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+132\zeta(3)\zeta(4)\right) \tag{628} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{5}(k+2)} &= \frac{1}{96}\left( -48 -96\zeta(2) -192\zeta(3) +480\zeta(4) +198\zeta(6) -192\zeta(3)^2 \right. \nonumber \\ &\left.+43\zeta(8) +120\zeta(2)\zeta(3)^2 -288\zeta(3)\zeta(5) +24 M(2,6)\right) \tag{629} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{6}(k+2)} &= \frac{-1}{96}\left( 96 +192\zeta(2) +384\zeta(3) -720\zeta(5) -96\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-198\zeta(6) +192\zeta(3)^2 -714\zeta(7) -192\zeta(2)\zeta(5) +792\zeta(3)\zeta(4) -43\zeta(8) \right. \nonumber \\ &\left.-120\zeta(2)\zeta(3)^2 +288\zeta(3)\zeta(5) -24 M(2,6) -788\zeta(9) +444\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+792\zeta(4)\zeta(5) -288\zeta(2)\zeta(7) -96\zeta(3)^3\right) \tag{630} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{5}(k+2)^{2}} &= \frac{-1}{768}\left( -156 -192\zeta(2) -264\zeta(3) -402\zeta(4) +1140\zeta(5) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3) -837\zeta(6) +360\zeta(3)^2 +2772\zeta(7) +384\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-2448\zeta(3)\zeta(4) +1190\zeta(8) +240\zeta(2)\zeta(3)^2 -1152\zeta(3)\zeta(5) -528 M(2,6)\right) \tag{631} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+1)(k+2)^{2}} &= \frac{-1}{64}\left( -28 -36\zeta(2) -52\zeta(3) +553\zeta(4) -410\zeta(5) \right. \nonumber \\ &\left.-40\zeta(2)\zeta(3) +186\zeta(6) -80\zeta(3)^2 -231\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+204\zeta(3)\zeta(4)\right) \tag{632} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)^{2}(k+2)^{2}} &= \frac{-1}{64}\left( -60 -80\zeta(2) -120\zeta(3) -854\zeta(4) +1020\zeta(5) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3) -93\zeta(6) +40\zeta(3)^2\right) \tag{633} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{3}(k+2)^{2}} &= \frac{-1}{16}\left( -32 -44\zeta(2) -68\zeta(3) +193\zeta(4) -10\zeta(5) \right. \nonumber \\ &\left.+33\zeta(6) -32\zeta(3)^2\right) \tag{634} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{4}(k+2)^{2}} &= \frac{1}{16}\left( 68 +96\zeta(2) +152\zeta(3) -26\zeta(4) -300\zeta(5) \right. \nonumber \\ &\left.-40\zeta(2)\zeta(3) -33\zeta(6) +32\zeta(3)^2 -119\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+132\zeta(3)\zeta(4)\right) \tag{635} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{5}(k+2)^{2}} &= \frac{-1}{96}\left( -864 -1248\zeta(2) -2016\zeta(3) +792\zeta(4) +3600\zeta(5) \right. \nonumber \\ &\left.+480\zeta(2)\zeta(3) +594\zeta(6) -576\zeta(3)^2 +1428\zeta(7) +384\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1584\zeta(3)\zeta(4) +43\zeta(8) +120\zeta(2)\zeta(3)^2 -288\zeta(3)\zeta(5) +24 M(2,6)\right) \tag{636} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+2)^{3}} &= \frac{-1}{128}\left( 164 +112\zeta(2) +72\zeta(3) -4\zeta(4) -324\zeta(5) \right. \nonumber \\ &\left.-64\zeta(2)\zeta(3) +156\zeta(6) -76\zeta(3)^2 -231\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+204\zeta(3)\zeta(4)\right) \tag{637} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)(k+2)^{3}} &= \frac{-1}{64}\left( 192 +148\zeta(2) +124\zeta(3) -557\zeta(4) +86\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(3) -30\zeta(6) +4\zeta(3)^2\right) \tag{638} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{2}(k+2)^{3}} &= \frac{1}{64}\left( -444 -376\zeta(2) -368\zeta(3) +260\zeta(4) +848\zeta(5) \right. \nonumber \\ &\left.+168\zeta(2)\zeta(3) -33\zeta(6) +32\zeta(3)^2\right) \tag{639} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{3}(k+2)^{3}} &= \frac{-1}{32}\left( 508 +464\zeta(2) +504\zeta(3) -646\zeta(4) -828\zeta(5) \right. \nonumber \\ &\left.-168\zeta(2)\zeta(3) -33\zeta(6) +32\zeta(3)^2\right) \tag{640} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{4}(k+2)^{3}} &= \frac{1}{16}\left( -576 -560\zeta(2) -656\zeta(3) +672\zeta(4) +1128\zeta(5) \right. \nonumber \\ &\left.+208\zeta(2)\zeta(3) +66\zeta(6) -64\zeta(3)^2 +119\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-132\zeta(3)\zeta(4)\right) \tag{641} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+2)^{4}} &= \frac{1}{128}\left( 676 +216\zeta(2) -96\zeta(3) -386\zeta(4) -256\zeta(5) \right. \nonumber \\ &\left.-112\zeta(2)\zeta(3) +22\zeta(6) -20\zeta(3)^2 -119\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+132\zeta(3)\zeta(4)\right) \tag{642} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)(k+2)^{4}} &= \frac{-1}{64}\left( -868 -364\zeta(2) -28\zeta(3) +943\zeta(4) +170\zeta(5) \right. \nonumber \\ &\left.+136\zeta(2)\zeta(3) +8\zeta(6) +16\zeta(3)^2 +119\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-132\zeta(3)\zeta(4)\right) \tag{643} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{2}(k+2)^{4}} &= \frac{-1}{64}\left( -2180 -1104\zeta(2) -424\zeta(3) +2146\zeta(4) +1188\zeta(5) \right. \nonumber \\ &\left.+440\zeta(2)\zeta(3) -17\zeta(6) +64\zeta(3)^2 +238\zeta(7) +64\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-264\zeta(3)\zeta(4)\right) \tag{644} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{3}(k+2)^{4}} &= \frac{1}{16}\left( 1344 +784\zeta(2) +464\zeta(3) -1396\zeta(4) -1008\zeta(5) \right. \nonumber \\ &\left.-304\zeta(2)\zeta(3) -8\zeta(6) -16\zeta(3)^2 -119\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+132\zeta(3)\zeta(4)\right) \tag{645} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+2)^{5}} &= \frac{1}{768}\left( -12444 -1224\zeta(2) +4320\zeta(3) +6144\zeta(4) +2232\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(3) +1911\zeta(6) -1152\zeta(3)^2 +852\zeta(7) +960\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1872\zeta(3)\zeta(4) -86\zeta(8) -240\zeta(2)\zeta(3)^2 +576\zeta(3)\zeta(5) -48 M(2,6)\right) \tag{646} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)(k+2)^{5}} &= \frac{-1}{384}\left( 17652 +3408\zeta(2) -4152\zeta(3) -11802\zeta(4) -3252\zeta(5) \right. \nonumber \\ &\left.-792\zeta(2)\zeta(3) -1959\zeta(6) +1056\zeta(3)^2 -1566\zeta(7) -1152\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+2664\zeta(3)\zeta(4) +86\zeta(8) +240\zeta(2)\zeta(3)^2 -576\zeta(3)\zeta(5) +48 M(2,6)\right) \tag{647} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{2}(k+2)^{5}} &= \frac{1}{96}\left( -12096 -3360\zeta(2) +1440\zeta(3) +9120\zeta(4) +3408\zeta(5) \right. \nonumber \\ &\left.+1056\zeta(2)\zeta(3) +954\zeta(6) -432\zeta(3)^2 +1140\zeta(7) +672\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1728\zeta(3)\zeta(4) -43\zeta(8) -120\zeta(2)\zeta(3)^2 +288\zeta(3)\zeta(5) -24 M(2,6)\right) \tag{648} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+2)^{6}} &= \frac{-1}{768}\left( -30732 +720\zeta(2) +11592\zeta(3) +10758\zeta(4) +8076\zeta(5) \right. \nonumber \\ &\left.-2712\zeta(2)\zeta(3) +5553\zeta(6) -3360\zeta(3)^2 +2442\zeta(7) +768\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-2808\zeta(3)\zeta(4) -4118\zeta(8) -240\zeta(2)\zeta(3)^2 +2880\zeta(3)\zeta(5) +1104 M(2,6) \right. \nonumber \\ &\left.+3152\zeta(9) -1776\zeta(3)\zeta(6) -3168\zeta(4)\zeta(5) +1152\zeta(2)\zeta(7) +384\zeta(3)^3\right) \tag{649} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)(k+2)^{6}} &= \frac{1}{96}\left( 12096 +672\zeta(2) -3936\zeta(3) -5640\zeta(4) -2832\zeta(5) \right. \nonumber \\ &\left.+480\zeta(2)\zeta(3) -1878\zeta(6) +1104\zeta(3)^2 -1002\zeta(7) -480\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+1368\zeta(3)\zeta(4) +1051\zeta(8) +120\zeta(2)\zeta(3)^2 -864\zeta(3)\zeta(5) -264 M(2,6) \right. \nonumber \\ &\left.-788\zeta(9) +444\zeta(3)\zeta(6) +792\zeta(4)\zeta(5) -288\zeta(2)\zeta(7) -96\zeta(3)^3\right) \tag{650} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+2)^{7}} &= \frac{1}{160}\left( -13440 +1120\zeta(2) +4640\zeta(3) +3520\zeta(4) +4080\zeta(5) \right. \nonumber \\ &\left.-1440\zeta(2)\zeta(3) +2300\zeta(6) -1200\zeta(3)^2 +2560\zeta(7) -480\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1200\zeta(3)\zeta(4) -1080\zeta(8) +480\zeta(3)\zeta(5) +480 M(2,6) +80\zeta(9) -720\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-240\zeta(4)\zeta(5) +480\zeta(2)\zeta(7) +160\zeta(3)^3 -501\zeta(10) +800\zeta(3)\zeta(7) \right. \nonumber \\ &\left.+80\zeta(3)^2\zeta(4) -560\zeta(2)\zeta(3)\zeta(5) +480\zeta(5)^2 +40 M(2,8)\right) \tag{651} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{6}} &= \frac{-1}{640}\left( 68823\zeta(10) -60000\zeta(3)\zeta(7) -1000\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+21680\zeta(2)\zeta(3)\zeta(5) -23560\zeta(5)^2 -12120 M(2,8) -1280\zeta(2) M(2,6)\right) \tag{652} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{5}(k+1)} &= \frac{1}{144}\left( 4320\zeta(5) +864\zeta(2)\zeta(3) -5874\zeta(6) -432\zeta(3)^2 \right. \nonumber \\ &\left.+3330\zeta(7) +720\zeta(2)\zeta(5) -3096\zeta(3)\zeta(4) +14833\zeta(8) +4032\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-16704\zeta(3)\zeta(5) -3744 M(2,6) +5232\zeta(9) -3348\zeta(3)\zeta(6) -3096\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+1008\zeta(2)\zeta(7) +480\zeta(3)^3\right) \tag{653} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{4}(k+1)^{2}} &= \frac{-1}{144}\left( 17280\zeta(5) +3456\zeta(2)\zeta(3) -22776\zeta(6) -1728\zeta(3)^2 \right. \nonumber \\ &\left.+6660\zeta(7) +1440\zeta(2)\zeta(5) -6192\zeta(3)\zeta(4) +14833\zeta(8) +4032\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-16704\zeta(3)\zeta(5) -3744 M(2,6)\right) \tag{654} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+1)^{3}} &= \frac{1}{4}\left( 720\zeta(5) +144\zeta(2)\zeta(3) -919\zeta(6) -72\zeta(3)^2 +147\zeta(7) \right. \nonumber \\ &\left.+40\zeta(2)\zeta(5) -160\zeta(3)\zeta(4)\right) \tag{655} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)^{4}} &= \frac{-1}{144}\left( 17280\zeta(5) +3456\zeta(2)\zeta(3) -21336\zeta(6) -1728\zeta(3)^2 \right. \nonumber \\ &\left.+3924\zeta(7) +1440\zeta(2)\zeta(5) -5328\zeta(3)\zeta(4) +12415\zeta(8) +3312\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-13824\zeta(3)\zeta(5) -3024 M(2,6)\right) \tag{656} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{5}} &= \frac{-1}{144}\left( -4320\zeta(5) -864\zeta(2)\zeta(3) +5154\zeta(6) +432\zeta(3)^2 \right. \nonumber \\ &\left.-1962\zeta(7) -720\zeta(2)\zeta(5) +2664\zeta(3)\zeta(4) -12415\zeta(8) -3312\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+13824\zeta(3)\zeta(5) +3024 M(2,6) -2088\zeta(9) +1188\zeta(3)\zeta(6) +2664\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-1008\zeta(2)\zeta(7) -384\zeta(3)^3\right) \tag{657} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{6}} &= \frac{-1}{640}\left( 48647\zeta(10) -42080\zeta(3)\zeta(7) +280\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+12720\zeta(2)\zeta(3)\zeta(5) -13320\zeta(5)^2 -7640 M(2,8) -1280\zeta(2) M(2,6)\right) \tag{658} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{5}(k+2)} &= \frac{1}{1152}\left( 36 +108\zeta(2) +396\zeta(3) +666\zeta(4) +1080\zeta(5) \right. \nonumber \\ &\left.+216\zeta(2)\zeta(3) -2937\zeta(6) -216\zeta(3)^2 +3330\zeta(7) +720\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-3096\zeta(3)\zeta(4) +29666\zeta(8) +8064\zeta(2)\zeta(3)^2 -33408\zeta(3)\zeta(5) -7488 M(2,6) \right. \nonumber \\ &\left.+20928\zeta(9) -13392\zeta(3)\zeta(6) -12384\zeta(4)\zeta(5) +4032\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+1920\zeta(3)^3\right) \tag{659} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{4}(k+1)(k+2)} &= \frac{1}{576}\left( 36 +108\zeta(2) +396\zeta(3) +666\zeta(4) -16200\zeta(5) \right. \nonumber \\ &\left.-3240\zeta(2)\zeta(3) +20559\zeta(6) +1512\zeta(3)^2 -9990\zeta(7) -2160\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+9288\zeta(3)\zeta(4) -29666\zeta(8) -8064\zeta(2)\zeta(3)^2 +33408\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+7488 M(2,6)\right) \tag{660} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+1)^{2}(k+2)} &= \frac{1}{32}\left( 4 +12\zeta(2) +44\zeta(3) +74\zeta(4) +2040\zeta(5) \right. \nonumber \\ &\left.+408\zeta(2)\zeta(3) -2777\zeta(6) -216\zeta(3)^2 +370\zeta(7) +80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-344\zeta(3)\zeta(4)\right) \tag{661} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)^{3}(k+2)} &= \frac{-1}{16}\left( -4 -12\zeta(2) -44\zeta(3) -74\zeta(4) +840\zeta(5) \right. \nonumber \\ &\left.+168\zeta(2)\zeta(3) -899\zeta(6) -72\zeta(3)^2 +218\zeta(7) +80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-296\zeta(3)\zeta(4)\right) \tag{662} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{4}(k+2)} &= \frac{-1}{144}\left( -72 -216\zeta(2) -792\zeta(3) -1332\zeta(4) -2160\zeta(5) \right. \nonumber \\ &\left.-432\zeta(2)\zeta(3) +5154\zeta(6) +432\zeta(3)^2 -12415\zeta(8) -3312\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+13824\zeta(3)\zeta(5) +3024 M(2,6)\right) \tag{663} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{5}(k+2)} &= \frac{1}{144}\left( 144 +432\zeta(2) +1584\zeta(3) +2664\zeta(4) -5154\zeta(6) \right. \nonumber \\ &\left.-432\zeta(3)^2 -1962\zeta(7) -720\zeta(2)\zeta(5) +2664\zeta(3)\zeta(4) +12415\zeta(8) \right. \nonumber \\ &\left.+3312\zeta(2)\zeta(3)^2 -13824\zeta(3)\zeta(5) -3024 M(2,6) -2088\zeta(9) +1188\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+2664\zeta(4)\zeta(5) -1008\zeta(2)\zeta(7) -384\zeta(3)^3\right) \tag{664} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{4}(k+2)^{2}} &= \frac{-1}{576}\left( 252 +504\zeta(2) +1440\zeta(3) +1404\zeta(4) +1080\zeta(5) \right. \nonumber \\ &\left.+288\zeta(2)\zeta(3) -5694\zeta(6) -432\zeta(3)^2 +3330\zeta(7) +720\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-3096\zeta(3)\zeta(4) +14833\zeta(8) +4032\zeta(2)\zeta(3)^2 -16704\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-3744 M(2,6)\right) \tag{665} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+1)(k+2)^{2}} &= \frac{-1}{64}\left( 60 +124\zeta(2) +364\zeta(3) +386\zeta(4) -1560\zeta(5) \right. \nonumber \\ &\left.-296\zeta(2)\zeta(3) +1019\zeta(6) +72\zeta(3)^2 -370\zeta(7) -80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+344\zeta(3)\zeta(4)\right) \tag{666} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)^{2}(k+2)^{2}} &= \frac{-1}{16}\left( 32 +68\zeta(2) +204\zeta(3) +230\zeta(4) +240\zeta(5) \right. \nonumber \\ &\left.+56\zeta(2)\zeta(3) -879\zeta(6) -72\zeta(3)^2\right) \tag{667} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{3}(k+2)^{2}} &= \frac{-1}{16}\left( 68 +148\zeta(2) +452\zeta(3) +534\zeta(4) -360\zeta(5) \right. \nonumber \\ &\left.-56\zeta(2)\zeta(3) -859\zeta(6) -72\zeta(3)^2 -218\zeta(7) -80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+296\zeta(3)\zeta(4)\right) \tag{668} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{4}(k+2)^{2}} &= \frac{-1}{144}\left( 1296 +2880\zeta(2) +8928\zeta(3) +10944\zeta(4) -4320\zeta(5) \right. \nonumber \\ &\left.-576\zeta(2)\zeta(3) -20616\zeta(6) -1728\zeta(3)^2 -3924\zeta(7) -1440\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+5328\zeta(3)\zeta(4) +12415\zeta(8) +3312\zeta(2)\zeta(3)^2 -13824\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-3024 M(2,6)\right) \tag{669} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+2)^{3}} &= \frac{1}{64}\left( 192 +244\zeta(2) +508\zeta(3) +122\zeta(4) -168\zeta(5) \right. \nonumber \\ &\left.-56\zeta(2)\zeta(3) -853\zeta(6) -136\zeta(3)^2 +294\zeta(7) +80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-320\zeta(3)\zeta(4)\right) \tag{670} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)(k+2)^{3}} &= \frac{1}{64}\left( 444 +612\zeta(2) +1380\zeta(3) +630\zeta(4) -1896\zeta(5) \right. \nonumber \\ &\left.-408\zeta(2)\zeta(3) -687\zeta(6) -200\zeta(3)^2 +218\zeta(7) +80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-296\zeta(3)\zeta(4)\right) \tag{671} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{2}(k+2)^{3}} &= \frac{-1}{32}\left( -508 -748\zeta(2) -1788\zeta(3) -1090\zeta(4) +1416\zeta(5) \right. \nonumber \\ &\left.+296\zeta(2)\zeta(3) +2445\zeta(6) +344\zeta(3)^2 -218\zeta(7) -80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+296\zeta(3)\zeta(4)\right) \tag{672} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{3}(k+2)^{3}} &= \frac{-1}{2}\left( -72 -112\zeta(2) -280\zeta(3) -203\zeta(4) +222\zeta(5) \right. \nonumber \\ &\left.+44\zeta(2)\zeta(3) +413\zeta(6) +52\zeta(3)^2\right) \tag{673} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+2)^{4}} &= \frac{1}{576}\left( -7812 -5976\zeta(2) -7920\zeta(3) +5724\zeta(4) +6984\zeta(5) \right. \nonumber \\ &\left.+3168\zeta(2)\zeta(3) +4290\zeta(6) +1872\zeta(3)^2 +2322\zeta(7) +432\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-2088\zeta(3)\zeta(4) -12415\zeta(8) -3312\zeta(2)\zeta(3)^2 +13824\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+3024 M(2,6)\right) \tag{674} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)(k+2)^{4}} &= \frac{1}{576}\left( -19620 -17460\zeta(2) -28260\zeta(3) +5778\zeta(4) +31032\zeta(5) \right. \nonumber \\ &\left.+10008\zeta(2)\zeta(3) +14763\zeta(6) +5544\zeta(3)^2 +2682\zeta(7) +144\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1512\zeta(3)\zeta(4) -24830\zeta(8) -6624\zeta(2)\zeta(3)^2 +27648\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+6048 M(2,6)\right) \tag{675} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{2}(k+2)^{4}} &= \frac{1}{144}\left( -12096 -12096\zeta(2) -22176\zeta(3) -2016\zeta(4) +21888\zeta(5) \right. \nonumber \\ &\left.+6336\zeta(2)\zeta(3) +18384\zeta(6) +4320\zeta(3)^2 +360\zeta(7) -288\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+576\zeta(3)\zeta(4) -12415\zeta(8) -3312\zeta(2)\zeta(3)^2 +13824\zeta(3)\zeta(5) +3024 M(2,6)\right) \tag{676} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+2)^{5}} &= \frac{-1}{1152}\left( -52956 -22860\zeta(2) -12060\zeta(3) +44622\zeta(4) +25992\zeta(5) \right. \nonumber \\ &\left.+10728\zeta(2)\zeta(3) +9309\zeta(6) -1512\zeta(3)^2 +20286\zeta(7) +9648\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-27576\zeta(3)\zeta(4) -25862\zeta(8) -9504\zeta(2)\zeta(3)^2 +34560\zeta(3)\zeta(5) +5472 M(2,6) \right. \nonumber \\ &\left.-8352\zeta(9) +4752\zeta(3)\zeta(6) +10656\zeta(4)\zeta(5) -4032\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-1536\zeta(3)^3\right) \tag{677} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)(k+2)^{5}} &= \frac{-1}{144}\left( -18144 -10080\zeta(2) -10080\zeta(3) +12600\zeta(4) +14256\zeta(5) \right. \nonumber \\ &\left.+5184\zeta(2)\zeta(3) +6018\zeta(6) +1008\zeta(3)^2 +5742\zeta(7) +2448\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-7272\zeta(3)\zeta(4) -12673\zeta(8) -4032\zeta(2)\zeta(3)^2 +15552\zeta(3)\zeta(5) +2880 M(2,6) \right. \nonumber \\ &\left.-2088\zeta(9) +1188\zeta(3)\zeta(6) +2664\zeta(4)\zeta(5) -1008\zeta(2)\zeta(7) -384\zeta(3)^3\right) \tag{678} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+2)^{6}} &= \frac{1}{1920}\left( -241920 -53760\zeta(2) +26880\zeta(3) +161280\zeta(4) +88320\zeta(5) \right. \nonumber \\ &\left.+7680\zeta(2)\zeta(3) +56640\zeta(6) -30720\zeta(3)^2 +57120\zeta(7) +30720\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-82560\zeta(3)\zeta(4) -43760\zeta(8) -9600\zeta(2)\zeta(3)^2 +46080\zeta(3)\zeta(5) +9600 M(2,6) \right. \nonumber \\ &\left.+63040\zeta(9) -35520\zeta(3)\zeta(6) -63360\zeta(4)\zeta(5) +23040\zeta(2)\zeta(7) +7680\zeta(3)^3 \right. \nonumber \\ &\left.-145941\zeta(10) +126240\zeta(3)\zeta(7) -840\zeta(3)^2\zeta(4) -38160\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+39960\zeta(5)^2 +22920 M(2,8) +3840\zeta(2) M(2,6)\right) \tag{679} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{5}} &= \frac{1}{256}\left( -64433\zeta(10) +57760\zeta(3)\zeta(7) -360\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.-20560\zeta(2)\zeta(3)\zeta(5) +22648\zeta(5)^2 +10920 M(2,8) +1280\zeta(2) M(2,6)\right) \tag{680} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{4}(k+1)} &= \frac{1}{288}\left( -51408\zeta(6) -6480\zeta(3)^2 +36918\zeta(7) +8208\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+9504\zeta(3)\zeta(4) +67811\zeta(8) +19080\zeta(2)\zeta(3)^2 -78768\zeta(3)\zeta(5) -16920 M(2,6) \right. \nonumber \\ &\left.+37768\zeta(9) -58740\zeta(3)\zeta(6) +19008\zeta(4)\zeta(5) +9540\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-1440\zeta(3)^3\right) \tag{681} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{3}(k+1)^{2}} &= \frac{-1}{288}\left( -154224\zeta(6) -19440\zeta(3)^2 +107226\zeta(7) +24624\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+28512\zeta(3)\zeta(4) +67811\zeta(8) +19080\zeta(2)\zeta(3)^2 -78768\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-16920 M(2,6)\right) \tag{682} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+1)^{3}} &= \frac{1}{288}\left( -154224\zeta(6) -19440\zeta(3)^2 +103698\zeta(7) +24624\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+28512\zeta(3)\zeta(4) +65621\zeta(8) +17640\zeta(2)\zeta(3)^2 -72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-15480 M(2,6)\right) \tag{683} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)^{4}} &= \frac{1}{288}\left( 51408\zeta(6) +6480\zeta(3)^2 -33390\zeta(7) -8208\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-9504\zeta(3)\zeta(4) -65621\zeta(8) -17640\zeta(2)\zeta(3)^2 +72432\zeta(3)\zeta(5) +15480 M(2,6) \right. \nonumber \\ &\left.-28480\zeta(9) +51540\zeta(3)\zeta(6) -19008\zeta(4)\zeta(5) -9540\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+1440\zeta(3)^3\right) \tag{684} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{5}} &= \frac{1}{256}\left( 49901\zeta(10) -43040\zeta(3)\zeta(7) -1080\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+13840\zeta(2)\zeta(3)\zeta(5) -13592\zeta(5)^2 -7560 M(2,8) -1280\zeta(2) M(2,6)\right) \tag{685} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{4}(k+2)} &= \frac{-1}{1152}\left( 72 +288\zeta(2) +1512\zeta(3) +4518\zeta(4) +5112\zeta(5) \right. \nonumber \\ &\left.+1080\zeta(2)\zeta(3) +12852\zeta(6) +1620\zeta(3)^2 -18459\zeta(7) -4104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-4752\zeta(3)\zeta(4) -67811\zeta(8) -19080\zeta(2)\zeta(3)^2 +78768\zeta(3)\zeta(5) +16920 M(2,6) \right. \nonumber \\ &\left.-75536\zeta(9) +117480\zeta(3)\zeta(6) -38016\zeta(4)\zeta(5) -19080\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+2880\zeta(3)^3\right) \tag{686} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{3}(k+1)(k+2)} &= \frac{-1}{576}\left( 72 +288\zeta(2) +1512\zeta(3) +4518\zeta(4) +5112\zeta(5) \right. \nonumber \\ &\left.+1080\zeta(2)\zeta(3) -89964\zeta(6) -11340\zeta(3)^2 +55377\zeta(7) +12312\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+14256\zeta(3)\zeta(4) +67811\zeta(8) +19080\zeta(2)\zeta(3)^2 -78768\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-16920 M(2,6)\right) \tag{687} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+1)^{2}(k+2)} &= \frac{-1}{32}\left( 8 +32\zeta(2) +168\zeta(3) +502\zeta(4) +568\zeta(5) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3) +7140\zeta(6) +900\zeta(3)^2 -5761\zeta(7) -1368\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1584\zeta(3)\zeta(4)\right) \tag{688} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)^{3}(k+2)} &= \frac{-1}{288}\left( 144 +576\zeta(2) +3024\zeta(3) +9036\zeta(4) +10224\zeta(5) \right. \nonumber \\ &\left.+2160\zeta(2)\zeta(3) -25704\zeta(6) -3240\zeta(3)^2 +65621\zeta(8) +17640\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-72432\zeta(3)\zeta(5) -15480 M(2,6)\right) \tag{689} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{4}(k+2)} &= \frac{1}{288}\left( -288 -1152\zeta(2) -6048\zeta(3) -18072\zeta(4) -20448\zeta(5) \right. \nonumber \\ &\left.-4320\zeta(2)\zeta(3) +33390\zeta(7) +8208\zeta(2)\zeta(5) +9504\zeta(3)\zeta(4) -65621\zeta(8) \right. \nonumber \\ &\left.-17640\zeta(2)\zeta(3)^2 +72432\zeta(3)\zeta(5) +15480 M(2,6) +28480\zeta(9) -51540\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+19008\zeta(4)\zeta(5) +9540\zeta(2)\zeta(7) -1440\zeta(3)^3\right) \tag{690} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{3}(k+2)^{2}} &= \frac{-1}{1152}\left( -1080 -3024\zeta(2) -12888\zeta(3) -27666\zeta(4) -14760\zeta(5) \right. \nonumber \\ &\left.-3960\zeta(2)\zeta(3) -12786\zeta(6) -2700\zeta(3)^2 +53613\zeta(7) +12312\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+14256\zeta(3)\zeta(4) +67811\zeta(8) +19080\zeta(2)\zeta(3)^2 -78768\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-16920 M(2,6)\right) \tag{691} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+1)(k+2)^{2}} &= \frac{1}{96}\left( 192 +552\zeta(2) +2400\zeta(3) +5364\zeta(4) +3312\zeta(5) \right. \nonumber \\ &\left.+840\zeta(2)\zeta(3) -12863\zeta(6) -1440\zeta(3)^2 +294\zeta(7)\right) \tag{692} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)^{2}(k+2)^{2}} &= \frac{-1}{96}\left( -408 -1200\zeta(2) -5304\zeta(3) -12234\zeta(4) -8328\zeta(5) \right. \nonumber \\ &\left.-2040\zeta(2)\zeta(3) +4306\zeta(6) +180\zeta(3)^2 +16695\zeta(7) +4104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+4752\zeta(3)\zeta(4)\right) \tag{693} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{3}(k+2)^{2}} &= \frac{-1}{288}\left( -2592 -7776\zeta(2) -34848\zeta(3) -82440\zeta(4) -60192\zeta(5) \right. \nonumber \\ &\left.-14400\zeta(2)\zeta(3) +51540\zeta(6) +4320\zeta(3)^2 +100170\zeta(7) +24624\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+28512\zeta(3)\zeta(4) -65621\zeta(8) -17640\zeta(2)\zeta(3)^2 +72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+15480 M(2,6)\right) \tag{694} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+2)^{3}} &= \frac{1}{1152}\left( -7992 -15264\zeta(2) -51480\zeta(3) -71874\zeta(4) +4248\zeta(5) \right. \nonumber \\ &\left.-360\zeta(2)\zeta(3) +58584\zeta(6) +9540\zeta(3)^2 +32229\zeta(7) +5112\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+40896\zeta(3)\zeta(4) +65621\zeta(8) +17640\zeta(2)\zeta(3)^2 -72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-15480 M(2,6)\right) \tag{695} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)(k+2)^{3}} &= \frac{-1}{576}\left( 9144 +18576\zeta(2) +65880\zeta(3) +104058\zeta(4) +15624\zeta(5) \right. \nonumber \\ &\left.+5400\zeta(2)\zeta(3) -135762\zeta(6) -18180\zeta(3)^2 -30465\zeta(7) -5112\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-40896\zeta(3)\zeta(4) -65621\zeta(8) -17640\zeta(2)\zeta(3)^2 +72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+15480 M(2,6)\right) \tag{696} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{2}(k+2)^{3}} &= \frac{1}{288}\left( -10368 -22176\zeta(2) -81792\zeta(3) -140760\zeta(4) -40608\zeta(5) \right. \nonumber \\ &\left.-11520\zeta(2)\zeta(3) +148680\zeta(6) +18720\zeta(3)^2 +80550\zeta(7) +17424\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+55152\zeta(3)\zeta(4) +65621\zeta(8) +17640\zeta(2)\zeta(3)^2 -72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-15480 M(2,6)\right) \tag{697} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+2)^{4}} &= \frac{1}{1152}\left( 39240 +49968\zeta(2) +129384\zeta(3) +94806\zeta(4) -82440\zeta(5) \right. \nonumber \\ &\left.-25560\zeta(2)\zeta(3) -146718\zeta(6) -36540\zeta(3)^2 -675\zeta(7) +5976\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-37152\zeta(3)\zeta(4) +182679\zeta(8) +48600\zeta(2)\zeta(3)^2 -204048\zeta(3)\zeta(5) -45000 M(2,6) \right. \nonumber \\ &\left.-56960\zeta(9) +103080\zeta(3)\zeta(6) -38016\zeta(4)\zeta(5) -19080\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+2880\zeta(3)^3\right) \tag{698} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)(k+2)^{4}} &= \frac{1}{288}\left( 24192 +34272\zeta(2) +97632\zeta(3) +99432\zeta(4) -33408\zeta(5) \right. \nonumber \\ &\left.-10080\zeta(2)\zeta(3) -141240\zeta(6) -27360\zeta(3)^2 -15570\zeta(7) +432\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-39024\zeta(3)\zeta(4) +58529\zeta(8) +15480\zeta(2)\zeta(3)^2 -65808\zeta(3)\zeta(5) -14760 M(2,6) \right. \nonumber \\ &\left.-28480\zeta(9) +51540\zeta(3)\zeta(6) -19008\zeta(4)\zeta(5) -9540\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+1440\zeta(3)^3\right) \tag{699} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+2)^{5}} &= \frac{1}{2304}\left( -290304 -241920\zeta(2) -460800\zeta(3) -48960\zeta(4) +414720\zeta(5) \right. \nonumber \\ &\left.+149760\zeta(2)\zeta(3) +384960\zeta(6) +97920\zeta(3)^2 +162720\zeta(7) +57600\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-178560\zeta(3)\zeta(4) -1003520\zeta(8) -293760\zeta(2)\zeta(3)^2 +1175040\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+236160 M(2,6) -167040\zeta(9) +95040\zeta(3)\zeta(6) +213120\zeta(4)\zeta(5) -80640\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-30720\zeta(3)^3 +449109\zeta(10) -387360\zeta(3)\zeta(7) -9720\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+124560\zeta(2)\zeta(3)\zeta(5) -122328\zeta(5)^2 -68040 M(2,8) -11520\zeta(2) M(2,6)\right) \tag{700} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{4}} &= \frac{-1}{128}\left( 271367\zeta(10) -176560\zeta(3)\zeta(7) +84648\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+400\zeta(2)\zeta(3)\zeta(5) -121688\zeta(5)^2 -34376 M(2,8) -15040\zeta(2) M(2,6)\right) \tag{701} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{3}(k+1)} &= \frac{-1}{24}\left( -15456\zeta(7) -3480\zeta(2)\zeta(5) -7128\zeta(3)\zeta(4) +17529\zeta(8) \right. \nonumber \\ &\left.-984\zeta(2)\zeta(3)^2 +11688\zeta(3)\zeta(5) +1368 M(2,6) -7474\zeta(9) +13122\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-6048\zeta(4)\zeta(5) -1953\zeta(2)\zeta(7) +544\zeta(3)^3\right) \tag{702} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{2}(k+1)^{2}} &= \frac{-1}{6}\left( 7728\zeta(7) +1740\zeta(2)\zeta(5) +3564\zeta(3)\zeta(4) -8639\zeta(8) \right. \nonumber \\ &\left.+477\zeta(2)\zeta(3)^2 -5754\zeta(3)\zeta(5) -669 M(2,6)\right) \tag{703} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+1)^{3}} &= \frac{1}{24}\left( 15456\zeta(7) +3480\zeta(2)\zeta(5) +7128\zeta(3)\zeta(4) -17027\zeta(8) \right. \nonumber \\ &\left.+924\zeta(2)\zeta(3)^2 -11328\zeta(3)\zeta(5) -1308 M(2,6) +6146\zeta(9) -12582\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+5832\zeta(4)\zeta(5) +1953\zeta(2)\zeta(7) -536\zeta(3)^3\right) \tag{704} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)^{4}} &= \frac{-1}{128}\left( 259945\zeta(10) -163568\zeta(3)\zeta(7) +81848\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.-5200\zeta(2)\zeta(3)\zeta(5) -113288\zeta(5)^2 -31576 M(2,8) -15040\zeta(2) M(2,6)\right) \tag{705} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{3}(k+2)} &= \frac{-1}{96}\left( -12 -60\zeta(2) -408\zeta(3) -1713\zeta(4) -3426\zeta(5) \right. \nonumber \\ &\left.-732\zeta(2)\zeta(3) -6291\zeta(6) -804\zeta(3)^2 -7728\zeta(7) -1740\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-3564\zeta(3)\zeta(4) +17529\zeta(8) -984\zeta(2)\zeta(3)^2 +11688\zeta(3)\zeta(5) +1368 M(2,6) \right. \nonumber \\ &\left.-14948\zeta(9) +26244\zeta(3)\zeta(6) -12096\zeta(4)\zeta(5) -3906\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+1088\zeta(3)^3\right) \tag{706} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{2}(k+1)(k+2)} &= \frac{1}{16}\left( 4 +20\zeta(2) +136\zeta(3) +571\zeta(4) +1142\zeta(5) \right. \nonumber \\ &\left.+244\zeta(2)\zeta(3) +2097\zeta(6) +268\zeta(3)^2 -7728\zeta(7) -1740\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-3564\zeta(3)\zeta(4) +5843\zeta(8) -328\zeta(2)\zeta(3)^2 +3896\zeta(3)\zeta(5) +456 M(2,6)\right) \tag{707} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+1)^{2}(k+2)} &= \frac{1}{24}\left( 12 +60\zeta(2) +408\zeta(3) +1713\zeta(4) +3426\zeta(5) \right. \nonumber \\ &\left.+732\zeta(2)\zeta(3) +6291\zeta(6) +804\zeta(3)^2 +7728\zeta(7) +1740\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+3564\zeta(3)\zeta(4) -17027\zeta(8) +924\zeta(2)\zeta(3)^2 -11328\zeta(3)\zeta(5) -1308 M(2,6)\right) \tag{708} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)^{3}(k+2)} &= \frac{1}{24}\left( 24 +120\zeta(2) +816\zeta(3) +3426\zeta(4) +6852\zeta(5) \right. \nonumber \\ &\left.+1464\zeta(2)\zeta(3) +12582\zeta(6) +1608\zeta(3)^2 -17027\zeta(8) +924\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-11328\zeta(3)\zeta(5) -1308 M(2,6) -6146\zeta(9) +12582\zeta(3)\zeta(6) -5832\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-1953\zeta(2)\zeta(7) +536\zeta(3)^3\right) \tag{709} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{2}(k+2)^{2}} &= \frac{1}{96}\left( -192 -696\zeta(2) -3936\zeta(3) -12714\zeta(4) -16956\zeta(5) \right. \nonumber \\ &\left.-3912\zeta(2)\zeta(3) -12279\zeta(6) -2136\zeta(3)^2 +1239\zeta(7) +624\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-2376\zeta(3)\zeta(4) +34556\zeta(8) -1908\zeta(2)\zeta(3)^2 +23016\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+2676 M(2,6)\right) \tag{710} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+1)(k+2)^{2}} &= \frac{1}{48}\left( -204 -756\zeta(2) -4344\zeta(3) -14427\zeta(4) -20382\zeta(5) \right. \nonumber \\ &\left.-4644\zeta(2)\zeta(3) -18570\zeta(6) -2940\zeta(3)^2 +24423\zeta(7) +5844\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+8316\zeta(3)\zeta(4) +17027\zeta(8) -924\zeta(2)\zeta(3)^2 +11328\zeta(3)\zeta(5) +1308 M(2,6)\right) \tag{711} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)^{2}(k+2)^{2}} &= \frac{1}{24}\left( -216 -816\zeta(2) -4752\zeta(3) -16140\zeta(4) -23808\zeta(5) \right. \nonumber \\ &\left.-5376\zeta(2)\zeta(3) -24861\zeta(6) -3744\zeta(3)^2 +16695\zeta(7) +4104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+4752\zeta(3)\zeta(4) +34054\zeta(8) -1848\zeta(2)\zeta(3)^2 +22656\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+2616 M(2,6)\right) \tag{712} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+2)^{3}} &= \frac{1}{96}\left( 1524 +3948\zeta(2) +18360\zeta(3) +44121\zeta(4) +34122\zeta(5) \right. \nonumber \\ &\left.+8364\zeta(2)\zeta(3) -18408\zeta(6) -1692\zeta(3)^2 -32547\zeta(7) -6972\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-24012\zeta(3)\zeta(4) -82648\zeta(8) -16716\zeta(2)\zeta(3)^2 +61104\zeta(3)\zeta(5) +14172 M(2,6) \right. \nonumber \\ &\left.+12292\zeta(9) -25164\zeta(3)\zeta(6) +11664\zeta(4)\zeta(5) +3906\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-1072\zeta(3)^3\right) \tag{713} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)(k+2)^{3}} &= \frac{-1}{48}\left( -1728 -4704\zeta(2) -22704\zeta(3) -58548\zeta(4) -54504\zeta(5) \right. \nonumber \\ &\left.-13008\zeta(2)\zeta(3) -162\zeta(6) -1248\zeta(3)^2 +56970\zeta(7) +12816\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+32328\zeta(3)\zeta(4) +99675\zeta(8) +15792\zeta(2)\zeta(3)^2 -49776\zeta(3)\zeta(5) -12864 M(2,6) \right. \nonumber \\ &\left.-12292\zeta(9) +25164\zeta(3)\zeta(6) -11664\zeta(4)\zeta(5) -3906\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+1072\zeta(3)^3\right) \tag{714} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+2)^{4}} &= \frac{-1}{384}\left( 32256 +59136\zeta(2) +224256\zeta(3) +386496\zeta(4) +122880\zeta(5) \right. \nonumber \\ &\left.+26880\zeta(2)\zeta(3) -378528\zeta(6) -66432\zeta(3)^2 -167280\zeta(7) -23424\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-225792\zeta(3)\zeta(4) -28368\zeta(8) -8640\zeta(2)\zeta(3)^2 +26496\zeta(3)\zeta(5) +2880 M(2,6) \right. \nonumber \\ &\left.-227840\zeta(9) +412320\zeta(3)\zeta(6) -152064\zeta(4)\zeta(5) -76320\zeta(2)\zeta(7) +11520\zeta(3)^3 \right. \nonumber \\ &\left.+779835\zeta(10) -490704\zeta(3)\zeta(7) +245544\zeta(3)^2\zeta(4) -15600\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-339864\zeta(5)^2 -94728 M(2,8) -45120\zeta(2) M(2,6)\right) \tag{715} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k^{3}} &= \frac{1}{2560}\left( -16614991\zeta(10) +10315520\zeta(3)\zeta(7) -5879160\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+705040\zeta(2)\zeta(3)\zeta(5) +7710760\zeta(5)^2 +2021880 M(2,8) +1008000\zeta(2) M(2,6)\right) \tag{716} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k^{2}(k+1)} &= \frac{1}{72}\left( -479096\zeta(8) -12096\zeta(2)\zeta(3)^2 -109620\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+276341\zeta(9) +88665\zeta(3)\zeta(6) +143163\zeta(4)\zeta(5) +59166\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+4032\zeta(3)^3\right) \tag{717} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k(k+1)^{2}} &= \frac{1}{72}\left( 479096\zeta(8) +12096\zeta(2)\zeta(3)^2 +109620\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-269402\zeta(9) -88665\zeta(3)\zeta(6) -141273\zeta(4)\zeta(5) -59166\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-4032\zeta(3)^3\right) \tag{718} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{7}}{(k+1)^{3}} &= \frac{-1}{2560}\left( -16597239\zeta(10) +9974400\zeta(3)\zeta(7) -5800760\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+834960\zeta(2)\zeta(3)\zeta(5) +7473640\zeta(5)^2 +1956920 M(2,8) +1008000\zeta(2) M(2,6)\right) \tag{719} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k^{2}(k+2)} &= \frac{-1}{576}\left( 144 +864\zeta(2) +7200\zeta(3) +38664\zeta(4) +108504\zeta(5) \right. \nonumber \\ &\left.+23184\zeta(2)\zeta(3) +352887\zeta(6) +45864\zeta(3)^2 +319554\zeta(7) +73080\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+148932\zeta(3)\zeta(4) +958192\zeta(8) +24192\zeta(2)\zeta(3)^2 +219240\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-1105364\zeta(9) -354660\zeta(3)\zeta(6) -572652\zeta(4)\zeta(5) -236664\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-16128\zeta(3)^3\right) \tag{720} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k(k+1)(k+2)} &= \frac{-1}{288}\left( 144 +864\zeta(2) +7200\zeta(3) +38664\zeta(4) +108504\zeta(5) \right. \nonumber \\ &\left.+23184\zeta(2)\zeta(3) +352887\zeta(6) +45864\zeta(3)^2 +319554\zeta(7) +73080\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+148932\zeta(3)\zeta(4) -958192\zeta(8) -24192\zeta(2)\zeta(3)^2 -219240\zeta(3)\zeta(5)\right) \tag{721} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{(k+1)^{2}(k+2)} &= \frac{1}{144}\left( -144 -864\zeta(2) -7200\zeta(3) -38664\zeta(4) -108504\zeta(5) \right. \nonumber \\ &\left.-23184\zeta(2)\zeta(3) -352887\zeta(6) -45864\zeta(3)^2 -319554\zeta(7) -73080\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-148932\zeta(3)\zeta(4) +538804\zeta(9) +177330\zeta(3)\zeta(6) +282546\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+118332\zeta(2)\zeta(7) +8064\zeta(3)^3\right) \tag{722} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k(k+2)^{2}} &= \frac{1}{576}\left( 2448 +10944\zeta(2) +77184\zeta(3) +331344\zeta(4) +683928\zeta(5) \right. \nonumber \\ &\left.+152208\zeta(2)\zeta(3) +1403655\zeta(6) +199080\zeta(3)^2 +257472\zeta(7) +46872\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+247212\zeta(3)\zeta(4) -472076\zeta(8) +101808\zeta(2)\zeta(3)^2 -732312\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-109872 M(2,6) -1077608\zeta(9) -354660\zeta(3)\zeta(6) -565092\zeta(4)\zeta(5) -236664\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-16128\zeta(3)^3\right) \tag{723} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{(k+1)(k+2)^{2}} &= \frac{-1}{144}\left( -1296 -5904\zeta(2) -42192\zeta(3) -185004\zeta(4) -396216\zeta(5) \right. \nonumber \\ &\left.-87696\zeta(2)\zeta(3) -878271\zeta(6) -122472\zeta(3)^2 -288513\zeta(7) -59976\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-198072\zeta(3)\zeta(4) +715134\zeta(8) -38808\zeta(2)\zeta(3)^2 +475776\zeta(3)\zeta(5) +54936 M(2,6) \right. \nonumber \\ &\left.+538804\zeta(9) +177330\zeta(3)\zeta(6) +282546\zeta(4)\zeta(5) +118332\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+8064\zeta(3)^3\right) \tag{724} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{(k+2)^{3}} &= \frac{-1}{7680}\left( 276480 +913920\zeta(2) +5429760\zeta(3) +18389760\zeta(4) \right. \nonumber \\ &\left.+26899200\zeta(5) +6182400\zeta(2)\zeta(3) +28153920\zeta(6) +4381440\zeta(3)^2 -16691520\zeta(7) \right. \nonumber \\ &\left.-3951360\zeta(2)\zeta(5) -7674240\zeta(3)\zeta(4) -74888240\zeta(8) -7808640\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+15187200\zeta(3)\zeta(5) +5738880 M(2,6) +13767040\zeta(9) -28183680\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+13063680\zeta(4)\zeta(5) +4374720\zeta(2)\zeta(7) -1200640\zeta(3)^3 -49791717\zeta(10) \right. \nonumber \\ &\left.+29923200\zeta(3)\zeta(7) -17402280\zeta(3)^2\zeta(4) +2504880\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+22420920\zeta(5)^2 +5870760 M(2,8) +3024000\zeta(2) M(2,6)\right) \tag{725} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{8}}{k^{2}} &= \frac{-1}{480}\left( -18741581\zeta(10) -6689520\zeta(3)\zeta(7) +524640\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.-1452480\zeta(2)\zeta(3)\zeta(5) -4247040\zeta(5)^2 -485280 M(2,8) -299520\zeta(2) M(2,6)\right) \tag{726} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{k(k+1)} &= \frac{-1}{6}\left( -166700\zeta(9) -88665\zeta(3)\zeta(6) -80400\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-35091\zeta(2)\zeta(7) -4032\zeta(3)^3\right) \tag{727} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{(k+1)^{2}} &= \frac{-1}{240}\left( -9295879\zeta(10) -3314520\zeta(3)\zeta(7) +258540\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.-733800\zeta(2)\zeta(3)\zeta(5) -2098980\zeta(5)^2 -238860 M(2,8) -149760\zeta(2) M(2,6)\right) \tag{728} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{k(k+2)} &= \frac{1}{144}\left( 72 +504\zeta(2) +4968\zeta(3) +32400\zeta(4) +116280\zeta(5) \right. \nonumber \\ &\left.+24768\zeta(2)\zeta(3) +530346\zeta(6) +69264\zeta(3)^2 +849654\zeta(7) +193104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+404280\zeta(3)\zeta(4) +1906367\zeta(8) +48384\zeta(2)\zeta(3)^2 +436896\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+2000400\zeta(9) +1063980\zeta(3)\zeta(6) +964800\zeta(4)\zeta(5) +421092\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+48384\zeta(3)^3\right) \tag{729} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{(k+1)(k+2)} &= \frac{1}{72}\left( 72 +504\zeta(2) +4968\zeta(3) +32400\zeta(4) +116280\zeta(5) \right. \nonumber \\ &\left.+24768\zeta(2)\zeta(3) +530346\zeta(6) +69264\zeta(3)^2 +849654\zeta(7) +193104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+404280\zeta(3)\zeta(4) +1906367\zeta(8) +48384\zeta(2)\zeta(3)^2 +436896\zeta(3)\zeta(5)\right) \tag{730} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{(k+2)^{2}} &= \frac{1}{720}\left( -6480 -34560\zeta(2) -292320\zeta(3) -1564560\zeta(4) -4348800\zeta(5) \right. \nonumber \\ &\left.-950400\zeta(2)\zeta(3) -14106480\zeta(6) -1926720\zeta(3)^2 -12318480\zeta(7) -2712960\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-6755040\zeta(3)\zeta(4) -4760990\zeta(8) -1260000\zeta(2)\zeta(3)^2 +5146560\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+1098720 M(2,6) +21552160\zeta(9) +7093200\zeta(3)\zeta(6) +11301840\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+4733280\zeta(2)\zeta(7) +322560\zeta(3)^3 +27887637\zeta(10) +9943560\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-775620\zeta(3)^2\zeta(4) +2201400\zeta(2)\zeta(3)\zeta(5) +6296940\zeta(5)^2 +716580 M(2,8) \right. \nonumber \\ &\left.+449280\zeta(2) M(2,6)\right) \tag{731} \end{align}\]

Formulas for order \(r = m + n + p + q = 11\): \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{10}} &= \left( 6\zeta(11) -\zeta(2)\zeta(9) -\zeta(3)\zeta(8) -\zeta(4)\zeta(7) -\zeta(5)\zeta(6)\right) \tag{732} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{9}(k+1)} &= \frac{1}{4}\left( 4\zeta(2) -8\zeta(3) +5\zeta(4) -12\zeta(5) +4\zeta(2)\zeta(3) +7\zeta(6) \right. \nonumber \\ &\left.-2\zeta(3)^2 -16\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4) +9\zeta(8) -4\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-20\zeta(9) +4\zeta(3)\zeta(6) +4\zeta(4)\zeta(5) +4\zeta(2)\zeta(7) +11\zeta(10) -4\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-2\zeta(5)^2\right) \tag{733} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{8}(k+1)^{2}} &= \frac{1}{2}\left( -16\zeta(2) +30\zeta(3) -15\zeta(4) +30\zeta(5) -10\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-14\zeta(6) +4\zeta(3)^2 +24\zeta(7) -6\zeta(2)\zeta(5) -6\zeta(3)\zeta(4) -9\zeta(8) \right. \nonumber \\ &\left.+4\zeta(3)\zeta(5) +10\zeta(9) -2\zeta(3)\zeta(6) -2\zeta(4)\zeta(5) -2\zeta(2)\zeta(7)\right) \tag{734} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{7}(k+1)^{3}} &= \frac{-1}{4}\left( -112\zeta(2) +196\zeta(3) -74\zeta(4) +120\zeta(5) -40\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-42\zeta(6) +12\zeta(3)^2 +48\zeta(7) -12\zeta(2)\zeta(5) -12\zeta(3)\zeta(4) -9\zeta(8) \right. \nonumber \\ &\left.+4\zeta(3)\zeta(5)\right) \tag{735} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+1)^{4}} &= \frac{1}{2}\left( -112\zeta(2) +182\zeta(3) -47\zeta(4) +64\zeta(5) -22\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-14\zeta(6) +4\zeta(3)^2 +8\zeta(7) -2\zeta(2)\zeta(5) -2\zeta(3)\zeta(4)\right) \tag{736} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)^{5}} &= -\left( -70\zeta(2) +105\zeta(3) -15\zeta(4) +25\zeta(5) -10\zeta(2)\zeta(3) -\zeta(6)\right) \tag{737} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{6}} &= \frac{1}{2}\left( -112\zeta(2) +154\zeta(3) -5\zeta(4) +46\zeta(5) -22\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+6\zeta(6) -4\zeta(3)^2 +6\zeta(7) -2\zeta(2)\zeta(5) -2\zeta(3)\zeta(4)\right) \tag{738} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{7}} &= \frac{-1}{4}\left( -112\zeta(2) +140\zeta(3) +10\zeta(4) +80\zeta(5) -40\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+18\zeta(6) -12\zeta(3)^2 +36\zeta(7) -12\zeta(2)\zeta(5) -12\zeta(3)\zeta(4) +5\zeta(8) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(5)\right) \tag{739} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{8}} &= \frac{1}{2}\left( -16\zeta(2) +18\zeta(3) +3\zeta(4) +20\zeta(5) -10\zeta(2)\zeta(3) +6\zeta(6) \right. \nonumber \\ &\left.-4\zeta(3)^2 +18\zeta(7) -6\zeta(2)\zeta(5) -6\zeta(3)\zeta(4) +5\zeta(8) -4\zeta(3)\zeta(5) +8\zeta(9) \right. \nonumber \\ &\left.-2\zeta(3)\zeta(6) -2\zeta(4)\zeta(5) -2\zeta(2)\zeta(7)\right) \tag{740} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{9}} &= \frac{1}{4}\left( 4\zeta(2) -4\zeta(3) -\zeta(4) -8\zeta(5) +4\zeta(2)\zeta(3) -3\zeta(6) \right. \nonumber \\ &\left.+2\zeta(3)^2 -12\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4) -5\zeta(8) +4\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-16\zeta(9) +4\zeta(3)\zeta(6) +4\zeta(4)\zeta(5) +4\zeta(2)\zeta(7) -7\zeta(10) +4\zeta(3)\zeta(7) \right. \nonumber \\ &\left.+2\zeta(5)^2\right) \tag{741} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{10}} &= -\left( -5\zeta(11) +\zeta(2)\zeta(9) +\zeta(3)\zeta(8) +\zeta(4)\zeta(7) +\zeta(5)\zeta(6)\right) \tag{742} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{9}(k+2)} &= \frac{1}{512}\left( 1 +\zeta(2) -4\zeta(3) +5\zeta(4) -24\zeta(5) +8\zeta(2)\zeta(3) +28\zeta(6) \right. \nonumber \\ &\left.-8\zeta(3)^2 -128\zeta(7) +32\zeta(2)\zeta(5) +32\zeta(3)\zeta(4) +144\zeta(8) -64\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-640\zeta(9) +128\zeta(3)\zeta(6) +128\zeta(4)\zeta(5) +128\zeta(2)\zeta(7) +704\zeta(10) \right. \nonumber \\ &\left.-256\zeta(3)\zeta(7) -128\zeta(5)^2\right) \tag{743} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{8}(k+1)(k+2)} &= \frac{-1}{256}\left( -1 +255\zeta(2) -508\zeta(3) +315\zeta(4) -744\zeta(5) \right. \nonumber \\ &\left.+248\zeta(2)\zeta(3) +420\zeta(6) -120\zeta(3)^2 -896\zeta(7) +224\zeta(2)\zeta(5) +224\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+432\zeta(8) -192\zeta(3)\zeta(5) -640\zeta(9) +128\zeta(3)\zeta(6) +128\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+128\zeta(2)\zeta(7)\right) \tag{744} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{7}(k+1)^{2}(k+2)} &= \frac{-1}{128}\left( -1 -769\zeta(2) +1412\zeta(3) -645\zeta(4) +1176\zeta(5) \right. \nonumber \\ &\left.-392\zeta(2)\zeta(3) -476\zeta(6) +136\zeta(3)^2 +640\zeta(7) -160\zeta(2)\zeta(5) -160\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-144\zeta(8) +64\zeta(3)\zeta(5)\right) \tag{745} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+1)^{3}(k+2)} &= \frac{1}{64}\left( 1 -1023\zeta(2) +1724\zeta(3) -539\zeta(4) +744\zeta(5) \right. \nonumber \\ &\left.-248\zeta(2)\zeta(3) -196\zeta(6) +56\zeta(3)^2 +128\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-32\zeta(3)\zeta(4)\right) \tag{746} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)^{4}(k+2)} &= \frac{1}{32}\left( 1 +769\zeta(2) -1188\zeta(3) +213\zeta(4) -280\zeta(5) \right. \nonumber \\ &\left.+104\zeta(2)\zeta(3) +28\zeta(6) -8\zeta(3)^2\right) \tag{747} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{5}(k+2)} &= \frac{-1}{16}\left( -1 +351\zeta(2) -492\zeta(3) +27\zeta(4) -120\zeta(5) \right. \nonumber \\ &\left.+56\zeta(2)\zeta(3) -12\zeta(6) +8\zeta(3)^2\right) \tag{748} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{6}(k+2)} &= \frac{1}{8}\left( 1 +97\zeta(2) -124\zeta(3) -7\zeta(4) -64\zeta(5) +32\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-12\zeta(6) +8\zeta(3)^2 -24\zeta(7) +8\zeta(2)\zeta(5) +8\zeta(3)\zeta(4)\right) \tag{749} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{7}(k+2)} &= \frac{1}{4}\left( 1 -15\zeta(2) +16\zeta(3) +3\zeta(4) +16\zeta(5) -8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+6\zeta(6) -4\zeta(3)^2 +12\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4) +5\zeta(8) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(5)\right) \tag{750} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{8}(k+2)} &= \frac{-1}{2}\left( -1 -\zeta(2) +2\zeta(3) +4\zeta(5) -2\zeta(2)\zeta(3) +6\zeta(7) \right. \nonumber \\ &\left.-2\zeta(2)\zeta(5) -2\zeta(3)\zeta(4) +8\zeta(9) -2\zeta(3)\zeta(6) -2\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-2\zeta(2)\zeta(7)\right) \tag{751} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{9}(k+2)} &= \frac{1}{4}\left( 4 -4\zeta(3) +\zeta(4) -8\zeta(5) +4\zeta(2)\zeta(3) +3\zeta(6) -2\zeta(3)^2 \right. \nonumber \\ &\left.-12\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4) +5\zeta(8) -4\zeta(3)\zeta(5) -16\zeta(9) \right. \nonumber \\ &\left.+4\zeta(3)\zeta(6) +4\zeta(4)\zeta(5) +4\zeta(2)\zeta(7) +7\zeta(10) -4\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-2\zeta(5)^2\right) \tag{752} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{8}(k+2)^{2}} &= \frac{-1}{256}\left( 6 +3\zeta(2) -15\zeta(3) +15\zeta(4) -60\zeta(5) +20\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+56\zeta(6) -16\zeta(3)^2 -192\zeta(7) +48\zeta(2)\zeta(5) +48\zeta(3)\zeta(4) +144\zeta(8) \right. \nonumber \\ &\left.-64\zeta(3)\zeta(5) -320\zeta(9) +64\zeta(3)\zeta(6) +64\zeta(4)\zeta(5) +64\zeta(2)\zeta(7)\right) \tag{753} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{7}(k+1)(k+2)^{2}} &= \frac{1}{256}\left( -13 +249\zeta(2) -478\zeta(3) +285\zeta(4) -624\zeta(5) \right. \nonumber \\ &\left.+208\zeta(2)\zeta(3) +308\zeta(6) -88\zeta(3)^2 -512\zeta(7) +128\zeta(2)\zeta(5) +128\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+144\zeta(8) -64\zeta(3)\zeta(5)\right) \tag{754} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+1)^{2}(k+2)^{2}} &= \frac{-1}{64}\left( 7 +260\zeta(2) -467\zeta(3) +180\zeta(4) -276\zeta(5) \right. \nonumber \\ &\left.+92\zeta(2)\zeta(3) +84\zeta(6) -24\zeta(3)^2 -64\zeta(7) +16\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+16\zeta(3)\zeta(4)\right) \tag{755} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)^{3}(k+2)^{2}} &= \frac{-1}{64}\left( 15 -503\zeta(2) +790\zeta(3) -179\zeta(4) +192\zeta(5) \right. \nonumber \\ &\left.-64\zeta(2)\zeta(3) -28\zeta(6) +8\zeta(3)^2\right) \tag{756} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{4}(k+2)^{2}} &= \frac{-1}{16}\left( 8 +133\zeta(2) -199\zeta(3) +17\zeta(4) -44\zeta(5) \right. \nonumber \\ &\left.+20\zeta(2)\zeta(3)\right) \tag{757} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{5}(k+2)^{2}} &= \frac{-1}{16}\left( 17 -85\zeta(2) +94\zeta(3) +7\zeta(4) +32\zeta(5) \right. \nonumber \\ &\left.-16\zeta(2)\zeta(3) +12\zeta(6) -8\zeta(3)^2\right) \tag{758} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{6}(k+2)^{2}} &= \frac{-1}{4}\left( 9 +6\zeta(2) -15\zeta(3) -16\zeta(5) +8\zeta(2)\zeta(3) -12\zeta(7) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4)\right) \tag{759} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{7}(k+2)^{2}} &= \frac{-1}{4}\left( 19 -3\zeta(2) -14\zeta(3) +3\zeta(4) -16\zeta(5) +8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+6\zeta(6) -4\zeta(3)^2 -12\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4) +5\zeta(8) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(5)\right) \tag{760} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{8}(k+2)^{2}} &= \frac{1}{2}\left( -20 +2\zeta(2) +16\zeta(3) -3\zeta(4) +20\zeta(5) -10\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-6\zeta(6) +4\zeta(3)^2 +18\zeta(7) -6\zeta(2)\zeta(5) -6\zeta(3)\zeta(4) -5\zeta(8) +4\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+8\zeta(9) -2\zeta(3)\zeta(6) -2\zeta(4)\zeta(5) -2\zeta(2)\zeta(7)\right) \tag{761} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{7}(k+2)^{3}} &= \frac{1}{256}\left( 34 +5\zeta(2) -51\zeta(3) +37\zeta(4) -120\zeta(5) +40\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+84\zeta(6) -24\zeta(3)^2 -192\zeta(7) +48\zeta(2)\zeta(5) +48\zeta(3)\zeta(4) +72\zeta(8) \right. \nonumber \\ &\left.-32\zeta(3)\zeta(5)\right) \tag{762} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+1)(k+2)^{3}} &= \frac{1}{256}\left( 81 -239\zeta(2) +376\zeta(3) -211\zeta(4) +384\zeta(5) \right. \nonumber \\ &\left.-128\zeta(2)\zeta(3) -140\zeta(6) +40\zeta(3)^2 +128\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-32\zeta(3)\zeta(4)\right) \tag{763} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)^{2}(k+2)^{3}} &= \frac{-1}{128}\left( -95 -281\zeta(2) +558\zeta(3) -149\zeta(4) +168\zeta(5) \right. \nonumber \\ &\left.-56\zeta(2)\zeta(3) -28\zeta(6) +8\zeta(3)^2\right) \tag{764} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{3}(k+2)^{3}} &= \frac{1}{32}\left( 55 -111\zeta(2) +116\zeta(3) -15\zeta(4) +12\zeta(5) \right. \nonumber \\ &\left.-4\zeta(2)\zeta(3)\right) \tag{765} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{4}(k+2)^{3}} &= \frac{-1}{16}\left( -63 -22\zeta(2) +83\zeta(3) -2\zeta(4) +32\zeta(5) \right. \nonumber \\ &\left.-16\zeta(2)\zeta(3)\right) \tag{766} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{5}(k+2)^{3}} &= \frac{1}{16}\left( 143 -41\zeta(2) -72\zeta(3) +11\zeta(4) -32\zeta(5) \right. \nonumber \\ &\left.+16\zeta(2)\zeta(3) +12\zeta(6) -8\zeta(3)^2\right) \tag{767} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{6}(k+2)^{3}} &= \frac{1}{8}\left( 161 -29\zeta(2) -102\zeta(3) +11\zeta(4) -64\zeta(5) +32\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+12\zeta(6) -8\zeta(3)^2 -24\zeta(7) +8\zeta(2)\zeta(5) +8\zeta(3)\zeta(4)\right) \tag{768} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{7}(k+2)^{3}} &= \frac{1}{4}\left( 180 -32\zeta(2) -116\zeta(3) +14\zeta(4) -80\zeta(5) +40\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+18\zeta(6) -12\zeta(3)^2 -36\zeta(7) +12\zeta(2)\zeta(5) +12\zeta(3)\zeta(4) +5\zeta(8) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(5)\right) \tag{769} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+2)^{4}} &= \frac{1}{256}\left( -122 +9\zeta(2) +107\zeta(3) -43\zeta(4) +128\zeta(5) -44\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-56\zeta(6) +16\zeta(3)^2 +64\zeta(7) -16\zeta(2)\zeta(5) -16\zeta(3)\zeta(4)\right) \tag{770} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)(k+2)^{4}} &= \frac{1}{256}\left( -325 +257\zeta(2) -162\zeta(3) +125\zeta(4) -128\zeta(5) \right. \nonumber \\ &\left.+40\zeta(2)\zeta(3) +28\zeta(6) -8\zeta(3)^2\right) \tag{771} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{2}(k+2)^{4}} &= \frac{-1}{32}\left( 105 +6\zeta(2) -99\zeta(3) +6\zeta(4) -10\zeta(5) \right. \nonumber \\ &\left.+4\zeta(2)\zeta(3)\right) \tag{772} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{3}(k+2)^{4}} &= \frac{1}{32}\left( -265 +99\zeta(2) +82\zeta(3) +3\zeta(4) +8\zeta(5) \right. \nonumber \\ &\left.-4\zeta(2)\zeta(3)\right) \tag{773} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{4}(k+2)^{4}} &= \frac{-1}{16}\left( 328 -77\zeta(2) -165\zeta(3) -\zeta(4) -40\zeta(5) \right. \nonumber \\ &\left.+20\zeta(2)\zeta(3)\right) \tag{774} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{5}(k+2)^{4}} &= \frac{1}{16}\left( -799 +195\zeta(2) +402\zeta(3) -9\zeta(4) +112\zeta(5) \right. \nonumber \\ &\left.-56\zeta(2)\zeta(3) -12\zeta(6) +8\zeta(3)^2\right) \tag{775} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{6}(k+2)^{4}} &= \frac{-1}{2}\left( 240 -56\zeta(2) -126\zeta(3) +5\zeta(4) -44\zeta(5) +22\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+6\zeta(6) -4\zeta(3)^2 -6\zeta(7) +2\zeta(2)\zeta(5) +2\zeta(3)\zeta(4)\right) \tag{776} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+2)^{5}} &= \frac{1}{256}\left( 315 -58\zeta(2) -163\zeta(3) +2\zeta(4) -108\zeta(5) +40\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+8\zeta(6)\right) \tag{777} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)(k+2)^{5}} &= \frac{-1}{256}\left( -955 +373\zeta(2) +164\zeta(3) +121\zeta(4) +88\zeta(5) \right. \nonumber \\ &\left.-40\zeta(2)\zeta(3) +12\zeta(6) -8\zeta(3)^2\right) \tag{778} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{2}(k+2)^{5}} &= \frac{-1}{128}\left( -1375 +349\zeta(2) +560\zeta(3) +97\zeta(4) +128\zeta(5) \right. \nonumber \\ &\left.-56\zeta(2)\zeta(3) +12\zeta(6) -8\zeta(3)^2\right) \tag{779} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{3}(k+2)^{5}} &= \frac{1}{64}\left( 1905 -547\zeta(2) -724\zeta(3) -103\zeta(4) -144\zeta(5) \right. \nonumber \\ &\left.+64\zeta(2)\zeta(3) -12\zeta(6) +8\zeta(3)^2\right) \tag{780} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{4}(k+2)^{5}} &= \frac{-1}{32}\left( -2561 +701\zeta(2) +1054\zeta(3) +105\zeta(4) +224\zeta(5) \right. \nonumber \\ &\left.-104\zeta(2)\zeta(3) +12\zeta(6) -8\zeta(3)^2\right) \tag{781} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{5}(k+2)^{5}} &= -\left( -210 +56\zeta(2) +91\zeta(3) +6\zeta(4) +21\zeta(5) -10\zeta(2)\zeta(3)\right) \tag{782} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+2)^{6}} &= \frac{1}{256}\left( -634 +135\zeta(2) +205\zeta(3) +83\zeta(4) +140\zeta(5) -44\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+40\zeta(6) -16\zeta(3)^2 +48\zeta(7) -16\zeta(2)\zeta(5) -16\zeta(3)\zeta(4)\right) \tag{783} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)(k+2)^{6}} &= \frac{1}{256}\left( -2223 +643\zeta(2) +574\zeta(3) +287\zeta(4) +368\zeta(5) \right. \nonumber \\ &\left.-128\zeta(2)\zeta(3) +92\zeta(6) -40\zeta(3)^2 +96\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-32\zeta(3)\zeta(4)\right) \tag{784} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{2}(k+2)^{6}} &= \frac{1}{64}\left( -1799 +496\zeta(2) +567\zeta(3) +192\zeta(4) +248\zeta(5) \right. \nonumber \\ &\left.-92\zeta(2)\zeta(3) +52\zeta(6) -24\zeta(3)^2 +48\zeta(7) -16\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-16\zeta(3)\zeta(4)\right) \tag{785} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{3}(k+2)^{6}} &= \frac{-1}{64}\left( 5503 -1539\zeta(2) -1858\zeta(3) -487\zeta(4) -640\zeta(5) \right. \nonumber \\ &\left.+248\zeta(2)\zeta(3) -116\zeta(6) +56\zeta(3)^2 -96\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+32\zeta(3)\zeta(4)\right) \tag{786} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{4}(k+2)^{6}} &= \frac{1}{2}\left( -504 +140\zeta(2) +182\zeta(3) +37\zeta(4) +54\zeta(5) -22\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+8\zeta(6) -4\zeta(3)^2 +6\zeta(7) -2\zeta(2)\zeta(5) -2\zeta(3)\zeta(4)\right) \tag{787} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+2)^{7}} &= \frac{-1}{256}\left( -1058 +205\zeta(2) +233\zeta(3) +173\zeta(4) +208\zeta(5) \right. \nonumber \\ &\left.-40\zeta(2)\zeta(3) +116\zeta(6) -24\zeta(3)^2 +176\zeta(7) -48\zeta(2)\zeta(5) -48\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+40\zeta(8) -32\zeta(3)\zeta(5)\right) \tag{788} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)(k+2)^{7}} &= \frac{-1}{256}\left( -4339 +1053\zeta(2) +1040\zeta(3) +633\zeta(4) +784\zeta(5) \right. \nonumber \\ &\left.-208\zeta(2)\zeta(3) +324\zeta(6) -88\zeta(3)^2 +448\zeta(7) -128\zeta(2)\zeta(5) -128\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+80\zeta(8) -64\zeta(3)\zeta(5)\right) \tag{789} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{2}(k+2)^{7}} &= \frac{-1}{128}\left( -7937 +2045\zeta(2) +2174\zeta(3) +1017\zeta(4) +1280\zeta(5) \right. \nonumber \\ &\left.-392\zeta(2)\zeta(3) +428\zeta(6) -136\zeta(3)^2 +544\zeta(7) -160\zeta(2)\zeta(5) -160\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+80\zeta(8) -64\zeta(3)\zeta(5)\right) \tag{790} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{3}(k+2)^{7}} &= \frac{-1}{4}\left( -840 +224\zeta(2) +252\zeta(3) +94\zeta(4) +120\zeta(5) \right. \nonumber \\ &\left.-40\zeta(2)\zeta(3) +34\zeta(6) -12\zeta(3)^2 +40\zeta(7) -12\zeta(2)\zeta(5) -12\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+5\zeta(8) -4\zeta(3)\zeta(5)\right) \tag{791} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+2)^{8}} &= \frac{1}{256}\left( -1542 +243\zeta(2) +249\zeta(3) +231\zeta(4) +248\zeta(5) \right. \nonumber \\ &\left.-20\zeta(2)\zeta(3) +200\zeta(6) -16\zeta(3)^2 +272\zeta(7) -48\zeta(2)\zeta(5) -48\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+144\zeta(8) -64\zeta(3)\zeta(5) +256\zeta(9) -64\zeta(3)\zeta(6) -64\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-64\zeta(2)\zeta(7)\right) \tag{792} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)(k+2)^{8}} &= \frac{-1}{256}\left( 7423 -1539\zeta(2) -1538\zeta(3) -1095\zeta(4) -1280\zeta(5) \right. \nonumber \\ &\left.+248\zeta(2)\zeta(3) -724\zeta(6) +120\zeta(3)^2 -992\zeta(7) +224\zeta(2)\zeta(5) +224\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-368\zeta(8) +192\zeta(3)\zeta(5) -512\zeta(9) +128\zeta(3)\zeta(6) +128\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+128\zeta(2)\zeta(7)\right) \tag{793} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{2}(k+2)^{8}} &= \frac{-1}{2}\left( 240 -56\zeta(2) -58\zeta(3) -33\zeta(4) -40\zeta(5) +10\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-18\zeta(6) +4\zeta(3)^2 -24\zeta(7) +6\zeta(2)\zeta(5) +6\zeta(3)\zeta(4) -7\zeta(8) \right. \nonumber \\ &\left.+4\zeta(3)\zeta(5) -8\zeta(9) +2\zeta(3)\zeta(6) +2\zeta(4)\zeta(5) +2\zeta(2)\zeta(7)\right) \tag{794} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+2)^{9}} &= \frac{1}{512}\left( 4097 -509\zeta(2) -510\zeta(3) -505\zeta(4) -512\zeta(5) +8\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-492\zeta(6) +8\zeta(3)^2 -544\zeta(7) +32\zeta(2)\zeta(5) +32\zeta(3)\zeta(4) -464\zeta(8) \right. \nonumber \\ &\left.+64\zeta(3)\zeta(5) -768\zeta(9) +128\zeta(3)\zeta(6) +128\zeta(4)\zeta(5) +128\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-448\zeta(10) +256\zeta(3)\zeta(7) +128\zeta(5)^2\right) \tag{795} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)(k+2)^{9}} &= \frac{-1}{4}\left( -180 +32\zeta(2) +32\zeta(3) +25\zeta(4) +28\zeta(5) -4\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+19\zeta(6) -2\zeta(3)^2 +24\zeta(7) -4\zeta(2)\zeta(5) -4\zeta(3)\zeta(4) +13\zeta(8) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(5) +20\zeta(9) -4\zeta(3)\zeta(6) -4\zeta(4)\zeta(5) -4\zeta(2)\zeta(7) +7\zeta(10) \right. \nonumber \\ &\left.-4\zeta(3)\zeta(7) -2\zeta(5)^2\right) \tag{796} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+2)^{10}} &= \left( -10 +\zeta(2) +\zeta(3) +\zeta(4) +\zeta(5) +\zeta(6) +\zeta(7) +\zeta(8) +\zeta(9) \right. \nonumber \\ &\left.+\zeta(10) +5\zeta(11) -\zeta(2)\zeta(9) -\zeta(3)\zeta(8) -\zeta(4)\zeta(7) -\zeta(5)\zeta(6)\right) \tag{797} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{9}} &= \frac{1}{2}\left( 26\zeta(11) -2\zeta(2)\zeta(9) -9\zeta(3)\zeta(8) -5\zeta(4)\zeta(7) \right. \nonumber \\ &\left.-7\zeta(5)\zeta(6) +2\zeta(3)^2\zeta(5)\right) \tag{798} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{8}(k+1)} &= \frac{-1}{24}\left( 72\zeta(3) -102\zeta(4) +84\zeta(5) -24\zeta(2)\zeta(3) -97\zeta(6) \right. \nonumber \\ &\left.+48\zeta(3)^2 +144\zeta(7) -24\zeta(2)\zeta(5) -60\zeta(3)\zeta(4) -24 M(2,6) +220\zeta(9) \right. \nonumber \\ &\left.-84\zeta(3)\zeta(6) -60\zeta(4)\zeta(5) -24\zeta(2)\zeta(7) +8\zeta(3)^3 -24 M(2,8)\right) \tag{799} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{7}(k+1)^{2}} &= \frac{1}{12}\left( 252\zeta(3) -339\zeta(4) +210\zeta(5) -60\zeta(2)\zeta(3) -194\zeta(6) \right. \nonumber \\ &\left.+96\zeta(3)^2 +216\zeta(7) -36\zeta(2)\zeta(5) -90\zeta(3)\zeta(4) -24 M(2,6) +110\zeta(9) \right. \nonumber \\ &\left.-42\zeta(3)\zeta(6) -30\zeta(4)\zeta(5) -12\zeta(2)\zeta(7) +4\zeta(3)^3\right) \tag{800} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{6}(k+1)^{3}} &= \frac{-1}{4}\left( 252\zeta(3) -321\zeta(4) +146\zeta(5) -44\zeta(2)\zeta(3) -97\zeta(6) \right. \nonumber \\ &\left.+48\zeta(3)^2 +72\zeta(7) -12\zeta(2)\zeta(5) -30\zeta(3)\zeta(4) -4 M(2,6)\right) \tag{801} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+1)^{4}} &= \frac{1}{24}\left( 2520\zeta(3) -3030\zeta(4) +1020\zeta(5) -360\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-425\zeta(6) +216\zeta(3)^2 +144\zeta(7) -24\zeta(2)\zeta(5) -60\zeta(3)\zeta(4)\right) \tag{802} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)^{5}} &= \frac{1}{24}\left( -2520\zeta(3) +2850\zeta(4) -780\zeta(5) +360\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+245\zeta(6) -144\zeta(3)^2 -24\zeta(7) +24\zeta(2)\zeta(5) -12\zeta(3)\zeta(4)\right) \tag{803} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{6}} &= \frac{-1}{4}\left( -252\zeta(3) +267\zeta(4) -74\zeta(5) +44\zeta(2)\zeta(3) +37\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2 -12\zeta(7) +12\zeta(2)\zeta(5) -6\zeta(3)\zeta(4) -14\zeta(8) +8\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+4 M(2,6)\right) \tag{804} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{7}} &= \frac{-1}{12}\left( 252\zeta(3) -249\zeta(4) +90\zeta(5) -60\zeta(2)\zeta(3) -74\zeta(6) \right. \nonumber \\ &\left.+48\zeta(3)^2 +36\zeta(7) -36\zeta(2)\zeta(5) +18\zeta(3)\zeta(4) +84\zeta(8) -48\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-24 M(2,6) -2\zeta(9) +18\zeta(3)\zeta(6) +6\zeta(4)\zeta(5) -12\zeta(2)\zeta(7) -4\zeta(3)^3\right) \tag{805} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{8}} &= \frac{1}{24}\left( 72\zeta(3) -66\zeta(4) +36\zeta(5) -24\zeta(2)\zeta(3) -37\zeta(6) \right. \nonumber \\ &\left.+24\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) +12\zeta(3)\zeta(4) +84\zeta(8) -48\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-24 M(2,6) -4\zeta(9) +36\zeta(3)\zeta(6) +12\zeta(4)\zeta(5) -24\zeta(2)\zeta(7) -8\zeta(3)^3 \right. \nonumber \\ &\left.+108\zeta(10) -48\zeta(3)\zeta(7) -24\zeta(5)^2 -24 M(2,8)\right) \tag{806} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{9}} &= \frac{1}{2}\left( 4\zeta(11) +2\zeta(2)\zeta(9) -5\zeta(3)\zeta(8) -\zeta(4)\zeta(7) \right. \nonumber \\ &\left.-3\zeta(5)\zeta(6) +2\zeta(3)^2\zeta(5)\right) \tag{807} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{8}(k+2)} &= \frac{-1}{1536}\left( 6 +6\zeta(2) +18\zeta(3) -51\zeta(4) +84\zeta(5) -24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-194\zeta(6) +96\zeta(3)^2 +576\zeta(7) -96\zeta(2)\zeta(5) -240\zeta(3)\zeta(4) -192 M(2,6) \right. \nonumber \\ &\left.+3520\zeta(9) -1344\zeta(3)\zeta(6) -960\zeta(4)\zeta(5) -384\zeta(2)\zeta(7) +128\zeta(3)^3 \right. \nonumber \\ &\left.-768 M(2,8)\right) \tag{808} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{7}(k+1)(k+2)} &= \frac{1}{768}\left( -6 -6\zeta(2) +2286\zeta(3) -3213\zeta(4) +2604\zeta(5) \right. \nonumber \\ &\left.-744\zeta(2)\zeta(3) -2910\zeta(6) +1440\zeta(3)^2 +4032\zeta(7) -672\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1680\zeta(3)\zeta(4) -576 M(2,6) +3520\zeta(9) -1344\zeta(3)\zeta(6) -960\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-384\zeta(2)\zeta(7) +128\zeta(3)^3\right) \tag{809} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{6}(k+1)^{2}(k+2)} &= \frac{1}{384}\left( -6 -6\zeta(2) -5778\zeta(3) +7635\zeta(4) -4116\zeta(5) \right. \nonumber \\ &\left.+1176\zeta(2)\zeta(3) +3298\zeta(6) -1632\zeta(3)^2 -2880\zeta(7) +480\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+1200\zeta(3)\zeta(4) +192 M(2,6)\right) \tag{810} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+1)^{3}(k+2)} &= \frac{-1}{192}\left( 6 +6\zeta(2) -6318\zeta(3) +7773\zeta(4) -2892\zeta(5) \right. \nonumber \\ &\left.+936\zeta(2)\zeta(3) +1358\zeta(6) -672\zeta(3)^2 -576\zeta(7) +96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+240\zeta(3)\zeta(4)\right) \tag{811} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)^{4}(k+2)} &= \frac{1}{32}\left( -2 -2\zeta(2) -1254\zeta(3) +1449\zeta(4) -396\zeta(5) \right. \nonumber \\ &\left.+168\zeta(2)\zeta(3) +114\zeta(6) -64\zeta(3)^2\right) \tag{812} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{5}(k+2)} &= \frac{1}{48}\left( -6 -6\zeta(2) +1278\zeta(3) -1353\zeta(4) +372\zeta(5) \right. \nonumber \\ &\left.-216\zeta(2)\zeta(3) -148\zeta(6) +96\zeta(3)^2 +48\zeta(7) -48\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+24\zeta(3)\zeta(4)\right) \tag{813} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{6}(k+2)} &= \frac{1}{24}\left( -6 -6\zeta(2) -234\zeta(3) +249\zeta(4) -72\zeta(5) \right. \nonumber \\ &\left.+48\zeta(2)\zeta(3) +74\zeta(6) -48\zeta(3)^2 -24\zeta(7) +24\zeta(2)\zeta(5) -12\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-84\zeta(8) +48\zeta(3)\zeta(5) +24 M(2,6)\right) \tag{814} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{7}(k+2)} &= \frac{1}{6}\left( -3 -3\zeta(2) +9\zeta(3) +9\zeta(5) -6\zeta(2)\zeta(3) +6\zeta(7) \right. \nonumber \\ &\left.-6\zeta(2)\zeta(5) +3\zeta(3)\zeta(4) -\zeta(9) +9\zeta(3)\zeta(6) +3\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-6\zeta(2)\zeta(7) -2\zeta(3)^3\right) \tag{815} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{8}(k+2)} &= \frac{1}{24}\left( -24 -24\zeta(2) +66\zeta(4) +36\zeta(5) -24\zeta(2)\zeta(3) +37\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) +12\zeta(3)\zeta(4) -84\zeta(8) +48\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+24 M(2,6) -4\zeta(9) +36\zeta(3)\zeta(6) +12\zeta(4)\zeta(5) -24\zeta(2)\zeta(7) -8\zeta(3)^3 \right. \nonumber \\ &\left.-108\zeta(10) +48\zeta(3)\zeta(7) +24\zeta(5)^2 +24 M(2,8)\right) \tag{816} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{7}(k+2)^{2}} &= \frac{1}{1536}\left( 78 +42\zeta(2) +102\zeta(3) -339\zeta(4) +420\zeta(5) \right. \nonumber \\ &\left.-120\zeta(2)\zeta(3) -776\zeta(6) +384\zeta(3)^2 +1728\zeta(7) -288\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-720\zeta(3)\zeta(4) -384 M(2,6) +3520\zeta(9) -1344\zeta(3)\zeta(6) -960\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-384\zeta(2)\zeta(7) +128\zeta(3)^3\right) \tag{817} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{6}(k+1)(k+2)^{2}} &= \frac{-1}{384}\left( -42 -24\zeta(2) +1092\zeta(3) -1437\zeta(4) +1092\zeta(5) \right. \nonumber \\ &\left.-312\zeta(2)\zeta(3) -1067\zeta(6) +528\zeta(3)^2 +1152\zeta(7) -192\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-480\zeta(3)\zeta(4) -96 M(2,6)\right) \tag{818} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+1)^{2}(k+2)^{2}} &= \frac{-1}{128}\left( -30 -18\zeta(2) -1198\zeta(3) +1587\zeta(4) -644\zeta(5) \right. \nonumber \\ &\left.+184\zeta(2)\zeta(3) +388\zeta(6) -192\zeta(3)^2 -192\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+80\zeta(3)\zeta(4)\right) \tag{819} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)^{3}(k+2)^{2}} &= \frac{-1}{96}\left( -48 -30\zeta(2) +1362\zeta(3) -1506\zeta(4) +480\zeta(5) \right. \nonumber \\ &\left.-192\zeta(2)\zeta(3) -97\zeta(6) +48\zeta(3)^2\right) \tag{820} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{4}(k+2)^{2}} &= \frac{1}{96}\left( 102 +66\zeta(2) +1038\zeta(3) -1335\zeta(4) +228\zeta(5) \right. \nonumber \\ &\left.-120\zeta(2)\zeta(3) -148\zeta(6) +96\zeta(3)^2\right) \tag{821} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{5}(k+2)^{2}} &= \frac{1}{8}\left( 18 +12\zeta(2) -40\zeta(3) +3\zeta(4) -24\zeta(5) \right. \nonumber \\ &\left.+16\zeta(2)\zeta(3) -8\zeta(7) +8\zeta(2)\zeta(5) -4\zeta(3)\zeta(4)\right) \tag{822} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{6}(k+2)^{2}} &= \frac{1}{24}\left( 114 +78\zeta(2) -6\zeta(3) -231\zeta(4) -72\zeta(5) \right. \nonumber \\ &\left.+48\zeta(2)\zeta(3) -74\zeta(6) +48\zeta(3)^2 -24\zeta(7) +24\zeta(2)\zeta(5) -12\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+84\zeta(8) -48\zeta(3)\zeta(5) -24 M(2,6)\right) \tag{823} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{7}(k+2)^{2}} &= \frac{1}{12}\left( 120 +84\zeta(2) -24\zeta(3) -231\zeta(4) -90\zeta(5) \right. \nonumber \\ &\left.+60\zeta(2)\zeta(3) -74\zeta(6) +48\zeta(3)^2 -36\zeta(7) +36\zeta(2)\zeta(5) -18\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+84\zeta(8) -48\zeta(3)\zeta(5) -24 M(2,6) +2\zeta(9) -18\zeta(3)\zeta(6) -6\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+12\zeta(2)\zeta(7) +4\zeta(3)^3\right) \tag{824} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{6}(k+2)^{3}} &= \frac{-1}{512}\left( 162 +34\zeta(2) +54\zeta(3) -325\zeta(4) +292\zeta(5) \right. \nonumber \\ &\left.-88\zeta(2)\zeta(3) -388\zeta(6) +192\zeta(3)^2 +576\zeta(7) -96\zeta(2)\zeta(5) -240\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-64 M(2,6)\right) \tag{825} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+1)(k+2)^{3}} &= \frac{-1}{768}\left( 570 +150\zeta(2) -2022\zeta(3) +1899\zeta(4) -1308\zeta(5) \right. \nonumber \\ &\left.+360\zeta(2)\zeta(3) +970\zeta(6) -480\zeta(3)^2 -576\zeta(7) +96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+240\zeta(3)\zeta(4)\right) \tag{826} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)^{2}(k+2)^{3}} &= \frac{1}{192}\left( -330 -102\zeta(2) -786\zeta(3) +1431\zeta(4) -312\zeta(5) \right. \nonumber \\ &\left.+96\zeta(2)\zeta(3) +97\zeta(6) -48\zeta(3)^2\right) \tag{827} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{3}(k+2)^{3}} &= \frac{1}{32}\left( -126 -44\zeta(2) +192\zeta(3) -25\zeta(4) +56\zeta(5) \right. \nonumber \\ &\left.-32\zeta(2)\zeta(3)\right) \tag{828} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{4}(k+2)^{3}} &= \frac{-1}{96}\left( 858 +330\zeta(2) -114\zeta(3) -1185\zeta(4) -108\zeta(5) \right. \nonumber \\ &\left.+72\zeta(2)\zeta(3) -148\zeta(6) +96\zeta(3)^2\right) \tag{829} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{5}(k+2)^{3}} &= \frac{1}{48}\left( -966 -402\zeta(2) +354\zeta(3) +1167\zeta(4) +252\zeta(5) \right. \nonumber \\ &\left.-168\zeta(2)\zeta(3) +148\zeta(6) -96\zeta(3)^2 +48\zeta(7) -48\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+24\zeta(3)\zeta(4)\right) \tag{830} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{6}(k+2)^{3}} &= \frac{1}{4}\left( -180 -80\zeta(2) +60\zeta(3) +233\zeta(4) +54\zeta(5) \right. \nonumber \\ &\left.-36\zeta(2)\zeta(3) +37\zeta(6) -24\zeta(3)^2 +12\zeta(7) -12\zeta(2)\zeta(5) +6\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-14\zeta(8) +8\zeta(3)\zeta(5) +4 M(2,6)\right) \tag{831} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+2)^{4}} &= \frac{1}{1536}\left( 1950 -6\zeta(2) -282\zeta(3) -1647\zeta(4) +828\zeta(5) \right. \nonumber \\ &\left.-264\zeta(2)\zeta(3) -850\zeta(6) +432\zeta(3)^2 +576\zeta(7) -96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-240\zeta(3)\zeta(4)\right) \tag{832} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)(k+2)^{4}} &= \frac{1}{64}\left( 210 +12\zeta(2) -192\zeta(3) +21\zeta(4) -40\zeta(5) \right. \nonumber \\ &\left.+8\zeta(2)\zeta(3) +10\zeta(6) -4\zeta(3)^2\right) \tag{833} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{2}(k+2)^{4}} &= \frac{-1}{192}\left( -1590 -174\zeta(2) +366\zeta(3) +1305\zeta(4) -72\zeta(5) \right. \nonumber \\ &\left.+48\zeta(2)\zeta(3) +37\zeta(6) -24\zeta(3)^2\right) \tag{834} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{3}(k+2)^{4}} &= \frac{1}{96}\left( 1968 +306\zeta(2) -942\zeta(3) -1230\zeta(4) -96\zeta(5) \right. \nonumber \\ &\left.+48\zeta(2)\zeta(3) -37\zeta(6) +24\zeta(3)^2\right) \tag{835} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{4}(k+2)^{4}} &= \frac{-1}{32}\left( -1598 -314\zeta(2) +666\zeta(3) +1215\zeta(4) +100\zeta(5) \right. \nonumber \\ &\left.-56\zeta(2)\zeta(3) +74\zeta(6) -48\zeta(3)^2\right) \tag{836} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{5}(k+2)^{4}} &= \frac{-1}{24}\left( -2880 -672\zeta(2) +1176\zeta(3) +2406\zeta(4) +276\zeta(5) \right. \nonumber \\ &\left.-168\zeta(2)\zeta(3) +185\zeta(6) -120\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+12\zeta(3)\zeta(4)\right) \tag{837} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+2)^{5}} &= \frac{-1}{1536}\left( 5730 -702\zeta(2) -1818\zeta(3) -2073\zeta(4) -468\zeta(5) \right. \nonumber \\ &\left.+216\zeta(2)\zeta(3) -634\zeta(6) +384\zeta(3)^2 +96\zeta(7) -96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+48\zeta(3)\zeta(4)\right) \tag{838} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)(k+2)^{5}} &= \frac{1}{768}\left( -8250 +558\zeta(2) +4122\zeta(3) +1821\zeta(4) +948\zeta(5) \right. \nonumber \\ &\left.-312\zeta(2)\zeta(3) +514\zeta(6) -336\zeta(3)^2 -96\zeta(7) +96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-48\zeta(3)\zeta(4)\right) \tag{839} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{2}(k+2)^{5}} &= \frac{-1}{128}\left( 3810 -70\zeta(2) -1618\zeta(3) -1477\zeta(4) -268\zeta(5) \right. \nonumber \\ &\left.+72\zeta(2)\zeta(3) -196\zeta(6) +128\zeta(3)^2 +32\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+16\zeta(3)\zeta(4)\right) \tag{840} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{3}(k+2)^{5}} &= \frac{-1}{192}\left( 15366 +402\zeta(2) -6738\zeta(3) -6891\zeta(4) -996\zeta(5) \right. \nonumber \\ &\left.+312\zeta(2)\zeta(3) -662\zeta(6) +432\zeta(3)^2 +96\zeta(7) -96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+48\zeta(3)\zeta(4)\right) \tag{841} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{4}(k+2)^{5}} &= \frac{-1}{24}\left( 5040 +336\zeta(2) -2184\zeta(3) -2634\zeta(4) -324\zeta(5) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3) -221\zeta(6) +144\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+12\zeta(3)\zeta(4)\right) \tag{842} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+2)^{6}} &= \frac{-1}{512}\left( -4446 +774\zeta(2) +1482\zeta(3) +915\zeta(4) +1100\zeta(5) \right. \nonumber \\ &\left.-424\zeta(2)\zeta(3) +452\zeta(6) -256\zeta(3)^2 +288\zeta(7) -32\zeta(2)\zeta(5) -176\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-224\zeta(8) +128\zeta(3)\zeta(5) +64 M(2,6)\right) \tag{843} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)(k+2)^{6}} &= \frac{-1}{384}\left( -10794 +1440\zeta(2) +4284\zeta(3) +2283\zeta(4) +2124\zeta(5) \right. \nonumber \\ &\left.-792\zeta(2)\zeta(3) +935\zeta(6) -552\zeta(3)^2 +384\zeta(7) -288\zeta(3)\zeta(4) -336\zeta(8) \right. \nonumber \\ &\left.+192\zeta(3)\zeta(5) +96 M(2,6)\right) \tag{844} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{2}(k+2)^{6}} &= \frac{-1}{384}\left( -33018 +3090\zeta(2) +13422\zeta(3) +8997\zeta(4) +5052\zeta(5) \right. \nonumber \\ &\left.-1800\zeta(2)\zeta(3) +2458\zeta(6) -1488\zeta(3)^2 +672\zeta(7) +96\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-624\zeta(3)\zeta(4) -672\zeta(8) +384\zeta(3)\zeta(5) +192 M(2,6)\right) \tag{845} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{3}(k+2)^{6}} &= \frac{-1}{4}\left( -1008 +56\zeta(2) +420\zeta(3) +331\zeta(4) +126\zeta(5) \right. \nonumber \\ &\left.-44\zeta(2)\zeta(3) +65\zeta(6) -40\zeta(3)^2 +12\zeta(7) +4\zeta(2)\zeta(5) -14\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-14\zeta(8) +8\zeta(3)\zeta(5) +4 M(2,6)\right) \tag{846} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+2)^{7}} &= \frac{1}{1536}\left( -26034 +4782\zeta(2) +7578\zeta(3) +4389\zeta(4) +7020\zeta(5) \right. \nonumber \\ &\left.-2376\zeta(2)\zeta(3) +3032\zeta(6) -1248\zeta(3)^2 +4704\zeta(7) -1248\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1680\zeta(3)\zeta(4) -384\zeta(8) +384 M(2,6) +64\zeta(9) -576\zeta(3)\zeta(6) -192\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+384\zeta(2)\zeta(7) +128\zeta(3)^3\right) \tag{847} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)(k+2)^{7}} &= \frac{1}{768}\left( -47622 +7662\zeta(2) +16146\zeta(3) +8955\zeta(4) +11268\zeta(5) \right. \nonumber \\ &\left.-3960\zeta(2)\zeta(3) +4902\zeta(6) -2352\zeta(3)^2 +5472\zeta(7) -1248\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-2256\zeta(3)\zeta(4) -1056\zeta(8) +384\zeta(3)\zeta(5) +576 M(2,6) +64\zeta(9) -576\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-192\zeta(4)\zeta(5) +384\zeta(2)\zeta(7) +128\zeta(3)^3\right) \tag{848} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{2}(k+2)^{7}} &= \frac{1}{12}\left( -2520 +336\zeta(2) +924\zeta(3) +561\zeta(4) +510\zeta(5) \right. \nonumber \\ &\left.-180\zeta(2)\zeta(3) +230\zeta(6) -120\zeta(3)^2 +192\zeta(7) -36\zeta(2)\zeta(5) -90\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-54\zeta(8) +24\zeta(3)\zeta(5) +24 M(2,6) +2\zeta(9) -18\zeta(3)\zeta(6) -6\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+12\zeta(2)\zeta(7) +4\zeta(3)^3\right) \tag{849} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+2)^{8}} &= \frac{-1}{1536}\left( -44538 +7698\zeta(2) +10734\zeta(3) +6981\zeta(4) +10620\zeta(5) \right. \nonumber \\ &\left.-2952\zeta(2)\zeta(3) +5498\zeta(6) -1488\zeta(3)^2 +9888\zeta(7) -2592\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-2736\zeta(3)\zeta(4) +2976\zeta(8) -1920\zeta(3)\zeta(5) +192 M(2,6) +6208\zeta(9) \right. \nonumber \\ &\left.-2112\zeta(3)\zeta(6) -1728\zeta(4)\zeta(5) -1152\zeta(2)\zeta(7) +128\zeta(3)^3 -3456\zeta(10) \right. \nonumber \\ &\left.+1536\zeta(3)\zeta(7) +768\zeta(5)^2 +768 M(2,8)\right) \tag{850} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)(k+2)^{8}} &= \frac{-1}{24}\left( -2880 +480\zeta(2) +840\zeta(3) +498\zeta(4) +684\zeta(5) \right. \nonumber \\ &\left.-216\zeta(2)\zeta(3) +325\zeta(6) -120\zeta(3)^2 +480\zeta(7) -120\zeta(2)\zeta(5) -156\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+60\zeta(8) -48\zeta(3)\zeta(5) +24 M(2,6) +196\zeta(9) -84\zeta(3)\zeta(6) -60\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(7) +8\zeta(3)^3 -108\zeta(10) +48\zeta(3)\zeta(7) +24\zeta(5)^2 +24 M(2,8)\right) \tag{851} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+2)^{9}} &= \frac{-1}{2}\left( 90 -14\zeta(2) -18\zeta(3) -13\zeta(4) -18\zeta(5) +4\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-11\zeta(6) +2\zeta(3)^2 -18\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4) -9\zeta(8) \right. \nonumber \\ &\left.+4\zeta(3)\zeta(5) -18\zeta(9) +4\zeta(3)\zeta(6) +4\zeta(4)\zeta(5) +4\zeta(2)\zeta(7) -7\zeta(10) \right. \nonumber \\ &\left.+4\zeta(3)\zeta(7) +2\zeta(5)^2 -4\zeta(11) -2\zeta(2)\zeta(9) +5\zeta(3)\zeta(8) +\zeta(4)\zeta(7) \right. \nonumber \\ &\left.+3\zeta(5)\zeta(6) -2\zeta(3)^2\zeta(5)\right) \tag{852} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{8}} &= -\left( - M(3,8)\right) \tag{853} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{7}(k+1)} &= \frac{-1}{480}\left( -4800\zeta(4) +4800\zeta(5) +480\zeta(2)\zeta(3) -2790\zeta(6) \right. \nonumber \\ &\left.+1200\zeta(3)^2 +6930\zeta(7) +960\zeta(2)\zeta(5) -6120\zeta(3)\zeta(4) +2975\zeta(8) \right. \nonumber \\ &\left.+600\zeta(2)\zeta(3)^2 -2880\zeta(3)\zeta(5) -1320 M(2,6) +10420\zeta(9) -5820\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-6120\zeta(4)\zeta(5) +1440\zeta(2)\zeta(7) +960\zeta(3)^3 +4983\zeta(10) -3840\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-240\zeta(3)^2\zeta(4) +1680\zeta(2)\zeta(3)\zeta(5) -2160\zeta(5)^2 -1560 M(2,8)\right) \tag{854} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{6}(k+1)^{2}} &= \frac{1}{48}\left( -2880\zeta(4) +2760\zeta(5) +288\zeta(2)\zeta(3) -1116\zeta(6) \right. \nonumber \\ &\left.+480\zeta(3)^2 +2079\zeta(7) +288\zeta(2)\zeta(5) -1836\zeta(3)\zeta(4) +595\zeta(8) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3)^2 -576\zeta(3)\zeta(5) -264 M(2,6) +1042\zeta(9) -582\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-612\zeta(4)\zeta(5) +144\zeta(2)\zeta(7) +96\zeta(3)^3\right) \tag{855} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{5}(k+1)^{3}} &= \frac{-1}{96}\left( -14400\zeta(4) +13200\zeta(5) +1440\zeta(2)\zeta(3) -3546\zeta(6) \right. \nonumber \\ &\left.+1632\zeta(3)^2 +4158\zeta(7) +576\zeta(2)\zeta(5) -3672\zeta(3)\zeta(4) +595\zeta(8) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3)^2 -576\zeta(3)\zeta(5) -264 M(2,6)\right) \tag{856} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+1)^{4}} &= \frac{1}{8}\left( -1600\zeta(4) +1400\zeta(5) +160\zeta(2)\zeta(3) -252\zeta(6) \right. \nonumber \\ &\left.+144\zeta(3)^2 +175\zeta(7) +32\zeta(2)\zeta(5) -168\zeta(3)\zeta(4)\right) \tag{857} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)^{5}} &= \frac{1}{96}\left( 14400\zeta(4) -12000\zeta(5) -1440\zeta(2)\zeta(3) +1746\zeta(6) \right. \nonumber \\ &\left.-1392\zeta(3)^2 -2142\zeta(7) -576\zeta(2)\zeta(5) +2376\zeta(3)\zeta(4) +43\zeta(8) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3)^2 -288\zeta(3)\zeta(5) +24 M(2,6)\right) \tag{858} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{6}} &= \frac{-1}{48}\left( 2880\zeta(4) -2280\zeta(5) -288\zeta(2)\zeta(3) +396\zeta(6) \right. \nonumber \\ &\left.-384\zeta(3)^2 -1071\zeta(7) -288\zeta(2)\zeta(5) +1188\zeta(3)\zeta(4) +43\zeta(8) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3)^2 -288\zeta(3)\zeta(5) +24 M(2,6) -394\zeta(9) +222\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+396\zeta(4)\zeta(5) -144\zeta(2)\zeta(7) -48\zeta(3)^3\right) \tag{859} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{7}} &= \frac{-1}{480}\left( -4800\zeta(4) +3600\zeta(5) +480\zeta(2)\zeta(3) -990\zeta(6) \right. \nonumber \\ &\left.+960\zeta(3)^2 +3570\zeta(7) +960\zeta(2)\zeta(5) -3960\zeta(3)\zeta(4) -215\zeta(8) \right. \nonumber \\ &\left.-600\zeta(2)\zeta(3)^2 +1440\zeta(3)\zeta(5) -120 M(2,6) +3940\zeta(9) -2220\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-3960\zeta(4)\zeta(5) +1440\zeta(2)\zeta(7) +480\zeta(3)^3 -1503\zeta(10) +2400\zeta(3)\zeta(7) \right. \nonumber \\ &\left.+240\zeta(3)^2\zeta(4) -1680\zeta(2)\zeta(3)\zeta(5) +1440\zeta(5)^2 +120 M(2,8)\right) \tag{860} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{8}} &= \frac{1}{2}\left( -44\zeta(11) +21\zeta(3)\zeta(8) +9\zeta(4)\zeta(7) +15\zeta(5)\zeta(6) \right. \nonumber \\ &\left.-6\zeta(3)^2\zeta(5) +2 M(3,8)\right) \tag{861} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{7}(k+2)} &= \frac{-1}{7680}\left( -60 -120\zeta(2) -240\zeta(3) -600\zeta(4) +1200\zeta(5) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3) -1395\zeta(6) +600\zeta(3)^2 +6930\zeta(7) +960\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-6120\zeta(3)\zeta(4) +5950\zeta(8) +1200\zeta(2)\zeta(3)^2 -5760\zeta(3)\zeta(5) -2640 M(2,6) \right. \nonumber \\ &\left.+41680\zeta(9) -23280\zeta(3)\zeta(6) -24480\zeta(4)\zeta(5) +5760\zeta(2)\zeta(7) +3840\zeta(3)^3 \right. \nonumber \\ &\left.+39864\zeta(10) -30720\zeta(3)\zeta(7) -1920\zeta(3)^2\zeta(4) +13440\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-17280\zeta(5)^2 -12480 M(2,8)\right) \tag{862} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{6}(k+1)(k+2)} &= \frac{1}{768}\left( 12 +24\zeta(2) +48\zeta(3) -7560\zeta(4) +7440\zeta(5) \right. \nonumber \\ &\left.+744\zeta(2)\zeta(3) -4185\zeta(6) +1800\zeta(3)^2 +9702\zeta(7) +1344\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-8568\zeta(3)\zeta(4) +3570\zeta(8) +720\zeta(2)\zeta(3)^2 -3456\zeta(3)\zeta(5) -1584 M(2,6) \right. \nonumber \\ &\left.+8336\zeta(9) -4656\zeta(3)\zeta(6) -4896\zeta(4)\zeta(5) +1152\zeta(2)\zeta(7) +768\zeta(3)^3\right) \tag{863} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{5}(k+1)^{2}(k+2)} &= \frac{-1}{384}\left( -12 -24\zeta(2) -48\zeta(3) -15480\zeta(4) +14640\zeta(5) \right. \nonumber \\ &\left.+1560\zeta(2)\zeta(3) -4743\zeta(6) +2040\zeta(3)^2 +6930\zeta(7) +960\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-6120\zeta(3)\zeta(4) +1190\zeta(8) +240\zeta(2)\zeta(3)^2 -1152\zeta(3)\zeta(5) -528 M(2,6)\right) \tag{864} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+1)^{3}(k+2)} &= \frac{-1}{64}\left( -4 -8\zeta(2) -16\zeta(3) +4440\zeta(4) -3920\zeta(5) \right. \nonumber \\ &\left.-440\zeta(2)\zeta(3) +783\zeta(6) -408\zeta(3)^2 -462\zeta(7) -64\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+408\zeta(3)\zeta(4)\right) \tag{865} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)^{4}(k+2)} &= \frac{1}{32}\left( 4 +8\zeta(2) +16\zeta(3) +1960\zeta(4) -1680\zeta(5) \right. \nonumber \\ &\left.-200\zeta(2)\zeta(3) +225\zeta(6) -168\zeta(3)^2 -238\zeta(7) -64\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+264\zeta(3)\zeta(4)\right) \tag{866} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{5}(k+2)} &= \frac{-1}{96}\left( -24 -48\zeta(2) -96\zeta(3) +2640\zeta(4) -1920\zeta(5) \right. \nonumber \\ &\left.-240\zeta(2)\zeta(3) +396\zeta(6) -384\zeta(3)^2 -714\zeta(7) -192\zeta(2)\zeta(5) +792\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+43\zeta(8) +120\zeta(2)\zeta(3)^2 -288\zeta(3)\zeta(5) +24 M(2,6)\right) \tag{867} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{6}(k+2)} &= \frac{1}{48}\left( 24 +48\zeta(2) +96\zeta(3) +240\zeta(4) -360\zeta(5) \right. \nonumber \\ &\left.-48\zeta(2)\zeta(3) -357\zeta(7) -96\zeta(2)\zeta(5) +396\zeta(3)\zeta(4) -394\zeta(9) \right. \nonumber \\ &\left.+222\zeta(3)\zeta(6) +396\zeta(4)\zeta(5) -144\zeta(2)\zeta(7) -48\zeta(3)^3\right) \tag{868} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{7}(k+2)} &= \frac{-1}{480}\left( -480 -960\zeta(2) -1920\zeta(3) +3600\zeta(5) +480\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+990\zeta(6) -960\zeta(3)^2 +3570\zeta(7) +960\zeta(2)\zeta(5) -3960\zeta(3)\zeta(4) +215\zeta(8) \right. \nonumber \\ &\left.+600\zeta(2)\zeta(3)^2 -1440\zeta(3)\zeta(5) +120 M(2,6) +3940\zeta(9) -2220\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-3960\zeta(4)\zeta(5) +1440\zeta(2)\zeta(7) +480\zeta(3)^3 +1503\zeta(10) -2400\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-240\zeta(3)^2\zeta(4) +1680\zeta(2)\zeta(3)\zeta(5) -1440\zeta(5)^2 -120 M(2,8)\right) \tag{869} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{6}(k+2)^{2}} &= \frac{-1}{768}\left( 84 +108\zeta(2) +156\zeta(3) +261\zeta(4) -690\zeta(5) \right. \nonumber \\ &\left.-72\zeta(2)\zeta(3) +558\zeta(6) -240\zeta(3)^2 -2079\zeta(7) -288\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+1836\zeta(3)\zeta(4) -1190\zeta(8) -240\zeta(2)\zeta(3)^2 +1152\zeta(3)\zeta(5) +528 M(2,6) \right. \nonumber \\ &\left.-4168\zeta(9) +2328\zeta(3)\zeta(6) +2448\zeta(4)\zeta(5) -576\zeta(2)\zeta(7) -384\zeta(3)^3\right) \tag{870} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{5}(k+1)(k+2)^{2}} &= \frac{-1}{768}\left( 180 +240\zeta(2) +360\zeta(3) -7038\zeta(4) +6060\zeta(5) \right. \nonumber \\ &\left.+600\zeta(2)\zeta(3) -3069\zeta(6) +1320\zeta(3)^2 +5544\zeta(7) +768\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-4896\zeta(3)\zeta(4) +1190\zeta(8) +240\zeta(2)\zeta(3)^2 -1152\zeta(3)\zeta(5) -528 M(2,6)\right) \tag{871} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+1)^{2}(k+2)^{2}} &= \frac{-1}{64}\left( 32 +44\zeta(2) +68\zeta(3) +1407\zeta(4) -1430\zeta(5) \right. \nonumber \\ &\left.-160\zeta(2)\zeta(3) +279\zeta(6) -120\zeta(3)^2 -231\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+204\zeta(3)\zeta(4)\right) \tag{872} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)^{3}(k+2)^{2}} &= \frac{1}{64}\left( -68 -96\zeta(2) -152\zeta(3) +1626\zeta(4) -1060\zeta(5) \right. \nonumber \\ &\left.-120\zeta(2)\zeta(3) +225\zeta(6) -168\zeta(3)^2\right) \tag{873} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{4}(k+2)^{2}} &= \frac{1}{16}\left( -36 -52\zeta(2) -84\zeta(3) -167\zeta(4) +310\zeta(5) \right. \nonumber \\ &\left.+40\zeta(2)\zeta(3) +119\zeta(7) +32\zeta(2)\zeta(5) -132\zeta(3)\zeta(4)\right) \tag{874} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{5}(k+2)^{2}} &= \frac{1}{96}\left( -456 -672\zeta(2) -1104\zeta(3) +636\zeta(4) +1800\zeta(5) \right. \nonumber \\ &\left.+240\zeta(2)\zeta(3) +396\zeta(6) -384\zeta(3)^2 +714\zeta(7) +192\zeta(2)\zeta(5) -792\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+43\zeta(8) +120\zeta(2)\zeta(3)^2 -288\zeta(3)\zeta(5) +24 M(2,6)\right) \tag{875} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{6}(k+2)^{2}} &= \frac{1}{48}\left( -480 -720\zeta(2) -1200\zeta(3) +396\zeta(4) +2160\zeta(5) \right. \nonumber \\ &\left.+288\zeta(2)\zeta(3) +396\zeta(6) -384\zeta(3)^2 +1071\zeta(7) +288\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-1188\zeta(3)\zeta(4) +43\zeta(8) +120\zeta(2)\zeta(3)^2 -288\zeta(3)\zeta(5) +24 M(2,6) +394\zeta(9) \right. \nonumber \\ &\left.-222\zeta(3)\zeta(6) -396\zeta(4)\zeta(5) +144\zeta(2)\zeta(7) +48\zeta(3)^3\right) \tag{876} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{5}(k+2)^{3}} &= \frac{1}{1536}\left( 1140 +864\zeta(2) +696\zeta(3) +378\zeta(4) -3084\zeta(5) \right. \nonumber \\ &\left.-504\zeta(2)\zeta(3) +1773\zeta(6) -816\zeta(3)^2 -4158\zeta(7) -576\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+3672\zeta(3)\zeta(4) -1190\zeta(8) -240\zeta(2)\zeta(3)^2 +1152\zeta(3)\zeta(5) +528 M(2,6)\right) \tag{877} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+1)(k+2)^{3}} &= \frac{1}{128}\left( 220 +184\zeta(2) +176\zeta(3) -1110\zeta(4) +496\zeta(5) \right. \nonumber \\ &\left.+16\zeta(2)\zeta(3) -216\zeta(6) +84\zeta(3)^2 +231\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-204\zeta(3)\zeta(4)\right) \tag{878} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)^{2}(k+2)^{3}} &= \frac{-1}{64}\left( -252 -228\zeta(2) -244\zeta(3) -297\zeta(4) +934\zeta(5) \right. \nonumber \\ &\left.+144\zeta(2)\zeta(3) -63\zeta(6) +36\zeta(3)^2\right) \tag{879} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{3}(k+2)^{3}} &= \frac{1}{64}\left( 572 +552\zeta(2) +640\zeta(3) -1032\zeta(4) -808\zeta(5) \right. \nonumber \\ &\left.-168\zeta(2)\zeta(3) -99\zeta(6) +96\zeta(3)^2\right) \tag{880} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{4}(k+2)^{3}} &= \frac{-1}{32}\left( -644 -656\zeta(2) -808\zeta(3) +698\zeta(4) +1428\zeta(5) \right. \nonumber \\ &\left.+248\zeta(2)\zeta(3) +99\zeta(6) -96\zeta(3)^2 +238\zeta(7) +64\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-264\zeta(3)\zeta(4)\right) \tag{881} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{5}(k+2)^{3}} &= \frac{1}{96}\left( 4320 +4608\zeta(2) +5952\zeta(3) -4824\zeta(4) -10368\zeta(5) \right. \nonumber \\ &\left.-1728\zeta(2)\zeta(3) -990\zeta(6) +960\zeta(3)^2 -2142\zeta(7) -576\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+2376\zeta(3)\zeta(4) -43\zeta(8) -120\zeta(2)\zeta(3)^2 +288\zeta(3)\zeta(5) -24 M(2,6)\right) \tag{882} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+2)^{4}} &= \frac{1}{128}\left( -420 -164\zeta(2) +12\zeta(3) +195\zeta(4) +290\zeta(5) \right. \nonumber \\ &\left.+88\zeta(2)\zeta(3) -89\zeta(6) +48\zeta(3)^2 +175\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-168\zeta(3)\zeta(4)\right) \tag{883} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)(k+2)^{4}} &= \frac{-1}{128}\left( 1060 +512\zeta(2) +152\zeta(3) -1500\zeta(4) -84\zeta(5) \right. \nonumber \\ &\left.-160\zeta(2)\zeta(3) -38\zeta(6) -12\zeta(3)^2 -119\zeta(7) -32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+132\zeta(3)\zeta(4)\right) \tag{884} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{2}(k+2)^{4}} &= \frac{1}{64}\left( -1312 -740\zeta(2) -396\zeta(3) +1203\zeta(4) +1018\zeta(5) \right. \nonumber \\ &\left.+304\zeta(2)\zeta(3) -25\zeta(6) +48\zeta(3)^2 +119\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-132\zeta(3)\zeta(4)\right) \tag{885} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{3}(k+2)^{4}} &= \frac{1}{64}\left( -3196 -2032\zeta(2) -1432\zeta(3) +3438\zeta(4) +2844\zeta(5) \right. \nonumber \\ &\left.+776\zeta(2)\zeta(3) +49\zeta(6) +238\zeta(7) +64\zeta(2)\zeta(5) -264\zeta(3)\zeta(4)\right) \tag{886} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{4}(k+2)^{4}} &= \frac{1}{8}\left( -960 -672\zeta(2) -560\zeta(3) +1034\zeta(4) +1068\zeta(5) \right. \nonumber \\ &\left.+256\zeta(2)\zeta(3) +37\zeta(6) -24\zeta(3)^2 +119\zeta(7) +32\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-132\zeta(3)\zeta(4)\right) \tag{887} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+2)^{5}} &= \frac{-1}{1536}\left( -16500 -2520\zeta(2) +4896\zeta(3) +8460\zeta(4) +3768\zeta(5) \right. \nonumber \\ &\left.+648\zeta(2)\zeta(3) +1779\zeta(6) -1032\zeta(3)^2 +1566\zeta(7) +1152\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-2664\zeta(3)\zeta(4) -86\zeta(8) -240\zeta(2)\zeta(3)^2 +576\zeta(3)\zeta(5) -48 M(2,6)\right) \tag{888} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)(k+2)^{5}} &= \frac{1}{768}\left( 22860 +5592\zeta(2) -3984\zeta(3) -17460\zeta(4) -4272\zeta(5) \right. \nonumber \\ &\left.-1608\zeta(2)\zeta(3) -2007\zeta(6) +960\zeta(3)^2 -2280\zeta(7) -1344\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+3456\zeta(3)\zeta(4) +86\zeta(8) +240\zeta(2)\zeta(3)^2 -576\zeta(3)\zeta(5) +48 M(2,6)\right) \tag{889} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{2}(k+2)^{5}} &= \frac{-1}{384}\left( -30732 -10032\zeta(2) +1608\zeta(3) +24678\zeta(4) +10380\zeta(5) \right. \nonumber \\ &\left.+3432\zeta(2)\zeta(3) +1857\zeta(6) -672\zeta(3)^2 +2994\zeta(7) +1536\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-4248\zeta(3)\zeta(4) -86\zeta(8) -240\zeta(2)\zeta(3)^2 +576\zeta(3)\zeta(5) -48 M(2,6)\right) \tag{890} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{3}(k+2)^{5}} &= \frac{-1}{96}\left( -20160 -8064\zeta(2) -1344\zeta(3) +17496\zeta(4) +9456\zeta(5) \right. \nonumber \\ &\left.+2880\zeta(2)\zeta(3) +1002\zeta(6) -336\zeta(3)^2 +1854\zeta(7) +864\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-2520\zeta(3)\zeta(4) -43\zeta(8) -120\zeta(2)\zeta(3)^2 +288\zeta(3)\zeta(5) -24 M(2,6)\right) \tag{891} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+2)^{6}} &= \frac{1}{768}\left( -21588 -252\zeta(2) +7956\zeta(3) +8451\zeta(4) +5154\zeta(5) \right. \nonumber \\ &\left.-1368\zeta(2)\zeta(3) +3732\zeta(6) -2256\zeta(3)^2 +1647\zeta(7) +864\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-2340\zeta(3)\zeta(4) -2102\zeta(8) -240\zeta(2)\zeta(3)^2 +1728\zeta(3)\zeta(5) +528 M(2,6) \right. \nonumber \\ &\left.+1576\zeta(9) -888\zeta(3)\zeta(6) -1584\zeta(4)\zeta(5) +576\zeta(2)\zeta(7) +192\zeta(3)^3\right) \tag{892} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)(k+2)^{6}} &= \frac{-1}{768}\left( 66036 +6096\zeta(2) -19896\zeta(3) -34362\zeta(4) -14580\zeta(5) \right. \nonumber \\ &\left.+1128\zeta(2)\zeta(3) -9471\zeta(6) +5472\zeta(3)^2 -5574\zeta(7) -3072\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+8136\zeta(3)\zeta(4) +4290\zeta(8) +720\zeta(2)\zeta(3)^2 -4032\zeta(3)\zeta(5) -1008 M(2,6) \right. \nonumber \\ &\left.-3152\zeta(9) +1776\zeta(3)\zeta(6) +3168\zeta(4)\zeta(5) -1152\zeta(2)\zeta(7) -384\zeta(3)^3\right) \tag{893} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{2}(k+2)^{6}} &= \frac{-1}{48}\left( 12096 +2016\zeta(2) -2688\zeta(3) -7380\zeta(4) -3120\zeta(5) \right. \nonumber \\ &\left.-288\zeta(2)\zeta(3) -1416\zeta(6) +768\zeta(3)^2 -1071\zeta(7) -576\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+1548\zeta(3)\zeta(4) +547\zeta(8) +120\zeta(2)\zeta(3)^2 -576\zeta(3)\zeta(5) -120 M(2,6) -394\zeta(9) \right. \nonumber \\ &\left.+222\zeta(3)\zeta(6) +396\zeta(4)\zeta(5) -144\zeta(2)\zeta(7) -48\zeta(3)^3\right) \tag{894} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+2)^{7}} &= \frac{1}{7680}\left( 476220 -30480\zeta(2) -169320\zeta(3) -138270\zeta(4) -138300\zeta(5) \right. \nonumber \\ &\left.+48120\zeta(2)\zeta(3) -82965\zeta(6) +45600\zeta(3)^2 -73650\zeta(7) +7680\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+42840\zeta(3)\zeta(4) +46510\zeta(8) +1200\zeta(2)\zeta(3)^2 -25920\zeta(3)\zeta(5) -17040 M(2,6) \right. \nonumber \\ &\left.-17680\zeta(9) +26160\zeta(3)\zeta(6) +21600\zeta(4)\zeta(5) -17280\zeta(2)\zeta(7) -5760\zeta(3)^3 \right. \nonumber \\ &\left.+12024\zeta(10) -19200\zeta(3)\zeta(7) -1920\zeta(3)^2\zeta(4) +13440\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-11520\zeta(5)^2 -960 M(2,8)\right) \tag{895} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)(k+2)^{7}} &= \frac{1}{480}\left( 100800 -33600\zeta(3) -38760\zeta(4) -26400\zeta(5) \right. \nonumber \\ &\left.+6720\zeta(2)\zeta(3) -16290\zeta(6) +9120\zeta(3)^2 -12690\zeta(7) -960\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+10440\zeta(3)\zeta(4) +8495\zeta(8) +600\zeta(2)\zeta(3)^2 -5760\zeta(3)\zeta(5) -2760 M(2,6) \right. \nonumber \\ &\left.-4180\zeta(9) +4380\zeta(3)\zeta(6) +4680\zeta(4)\zeta(5) -2880\zeta(2)\zeta(7) -960\zeta(3)^3 \right. \nonumber \\ &\left.+1503\zeta(10) -2400\zeta(3)\zeta(7) -240\zeta(3)^2\zeta(4) +1680\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-1440\zeta(5)^2 -120 M(2,8)\right) \tag{896} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+2)^{8}} &= \frac{1}{8}\left( -960 +96\zeta(2) +304\zeta(3) +222\zeta(4) +284\zeta(5) -96\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+157\zeta(6) -72\zeta(3)^2 +216\zeta(7) -48\zeta(2)\zeta(5) -84\zeta(3)\zeta(4) -16\zeta(8) +24 M(2,6) \right. \nonumber \\ &\left.+100\zeta(9) -60\zeta(3)\zeta(6) -36\zeta(4)\zeta(5) +8\zeta(3)^3 -108\zeta(10) +48\zeta(3)\zeta(7) \right. \nonumber \\ &\left.+24\zeta(5)^2 +24 M(2,8) -176\zeta(11) +84\zeta(3)\zeta(8) +36\zeta(4)\zeta(7) +60\zeta(5)\zeta(6) \right. \nonumber \\ &\left.-24\zeta(3)^2\zeta(5) +8 M(3,8)\right) \tag{897} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{7}} &= \frac{-1}{48}\left( 2877\zeta(11) +272\zeta(2)\zeta(9) -1190\zeta(3)\zeta(8) -1212\zeta(4)\zeta(7) \right. \nonumber \\ &\left.-1018\zeta(5)\zeta(6) -80\zeta(2)\zeta(3)^3 +576\zeta(3)^2\zeta(5) -176 M(3,8)\right) \tag{898} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{6}(k+1)} &= \frac{1}{5760}\left( -172800\zeta(5) -34560\zeta(2)\zeta(3) +234960\zeta(6) +17280\zeta(3)^2 \right. \nonumber \\ &\left.-133200\zeta(7) -28800\zeta(2)\zeta(5) +123840\zeta(3)\zeta(4) -593320\zeta(8) \right. \nonumber \\ &\left.-161280\zeta(2)\zeta(3)^2 +668160\zeta(3)\zeta(5) +149760 M(2,6) -209280\zeta(9) \right. \nonumber \\ &\left.+133920\zeta(3)\zeta(6) +123840\zeta(4)\zeta(5) -40320\zeta(2)\zeta(7) -19200\zeta(3)^3 \right. \nonumber \\ &\left.-619407\zeta(10) +540000\zeta(3)\zeta(7) +9000\zeta(3)^2\zeta(4) -195120\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+212040\zeta(5)^2 +109080 M(2,8) +11520\zeta(2) M(2,6)\right) \tag{899} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{5}(k+1)^{2}} &= \frac{1}{72}\left( 10800\zeta(5) +2160\zeta(2)\zeta(3) -14325\zeta(6) -1080\zeta(3)^2 \right. \nonumber \\ &\left.+4995\zeta(7) +1080\zeta(2)\zeta(5) -4644\zeta(3)\zeta(4) +14833\zeta(8) +4032\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-16704\zeta(3)\zeta(5) -3744 M(2,6) +2616\zeta(9) -1674\zeta(3)\zeta(6) -1548\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+504\zeta(2)\zeta(7) +240\zeta(3)^3\right) \tag{900} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{4}(k+1)^{3}} &= \frac{1}{144}\left( -43200\zeta(5) -8640\zeta(2)\zeta(3) +55860\zeta(6) +4320\zeta(3)^2 \right. \nonumber \\ &\left.-11952\zeta(7) -2880\zeta(2)\zeta(5) +11952\zeta(3)\zeta(4) -14833\zeta(8) -4032\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+16704\zeta(3)\zeta(5) +3744 M(2,6)\right) \tag{901} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+1)^{4}} &= \frac{-1}{144}\left( -43200\zeta(5) -8640\zeta(2)\zeta(3) +54420\zeta(6) +4320\zeta(3)^2 \right. \nonumber \\ &\left.-9216\zeta(7) -2880\zeta(2)\zeta(5) +11088\zeta(3)\zeta(4) -12415\zeta(8) -3312\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+13824\zeta(3)\zeta(5) +3024 M(2,6)\right) \tag{902} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)^{5}} &= \frac{1}{72}\left( -10800\zeta(5) -2160\zeta(2)\zeta(3) +13245\zeta(6) +1080\zeta(3)^2 \right. \nonumber \\ &\left.-2943\zeta(7) -1080\zeta(2)\zeta(5) +3996\zeta(3)\zeta(4) -12415\zeta(8) -3312\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+13824\zeta(3)\zeta(5) +3024 M(2,6) -1044\zeta(9) +594\zeta(3)\zeta(6) +1332\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-504\zeta(2)\zeta(7) -192\zeta(3)^3\right) \tag{903} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{6}} &= \frac{-1}{5760}\left( -172800\zeta(5) -34560\zeta(2)\zeta(3) +206160\zeta(6) +17280\zeta(3)^2 \right. \nonumber \\ &\left.-78480\zeta(7) -28800\zeta(2)\zeta(5) +106560\zeta(3)\zeta(4) -496600\zeta(8) -132480\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+552960\zeta(3)\zeta(5) +120960 M(2,6) -83520\zeta(9) +47520\zeta(3)\zeta(6) +106560\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-40320\zeta(2)\zeta(7) -15360\zeta(3)^3 -437823\zeta(10) +378720\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-2520\zeta(3)^2\zeta(4) -114480\zeta(2)\zeta(3)\zeta(5) +119880\zeta(5)^2 +68760 M(2,8) \right. \nonumber \\ &\left.+11520\zeta(2) M(2,6)\right) \tag{904} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{7}} &= \frac{1}{48}\left( -237\zeta(11) -368\zeta(2)\zeta(9) +86\zeta(3)\zeta(8) +684\zeta(4)\zeta(7) \right. \nonumber \\ &\left.+202\zeta(5)\zeta(6) +80\zeta(2)\zeta(3)^3 -288\zeta(3)^2\zeta(5) -16 M(3,8)\right) \tag{905} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{6}(k+2)} &= \frac{1}{11520}\left( -180 -540\zeta(2) -1980\zeta(3) -3330\zeta(4) -5400\zeta(5) \right. \nonumber \\ &\left.-1080\zeta(2)\zeta(3) +14685\zeta(6) +1080\zeta(3)^2 -16650\zeta(7) -3600\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+15480\zeta(3)\zeta(4) -148330\zeta(8) -40320\zeta(2)\zeta(3)^2 +167040\zeta(3)\zeta(5) +37440 M(2,6) \right. \nonumber \\ &\left.-104640\zeta(9) +66960\zeta(3)\zeta(6) +61920\zeta(4)\zeta(5) -20160\zeta(2)\zeta(7) -9600\zeta(3)^3 \right. \nonumber \\ &\left.-619407\zeta(10) +540000\zeta(3)\zeta(7) +9000\zeta(3)^2\zeta(4) -195120\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+212040\zeta(5)^2 +109080 M(2,8) +11520\zeta(2) M(2,6)\right) \tag{906} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{5}(k+1)(k+2)} &= \frac{1}{384}\left( -12 -36\zeta(2) -132\zeta(3) -222\zeta(4) +11160\zeta(5) \right. \nonumber \\ &\left.+2232\zeta(2)\zeta(3) -14685\zeta(6) -1080\zeta(3)^2 +7770\zeta(7) +1680\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-7224\zeta(3)\zeta(4) +29666\zeta(8) +8064\zeta(2)\zeta(3)^2 -33408\zeta(3)\zeta(5) -7488 M(2,6) \right. \nonumber \\ &\left.+6976\zeta(9) -4464\zeta(3)\zeta(6) -4128\zeta(4)\zeta(5) +1344\zeta(2)\zeta(7) +640\zeta(3)^3\right) \tag{907} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{4}(k+1)^{2}(k+2)} &= \frac{-1}{576}\left( 36 +108\zeta(2) +396\zeta(3) +666\zeta(4) +52920\zeta(5) \right. \nonumber \\ &\left.+10584\zeta(2)\zeta(3) -70545\zeta(6) -5400\zeta(3)^2 +16650\zeta(7) +3600\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-15480\zeta(3)\zeta(4) +29666\zeta(8) +8064\zeta(2)\zeta(3)^2 -33408\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-7488 M(2,6)\right) \tag{908} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+1)^{3}(k+2)} &= \frac{-1}{32}\left( 4 +12\zeta(2) +44\zeta(3) +74\zeta(4) -3720\zeta(5) \right. \nonumber \\ &\left.-744\zeta(2)\zeta(3) +4575\zeta(6) +360\zeta(3)^2 -806\zeta(7) -240\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+936\zeta(3)\zeta(4)\right) \tag{909} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)^{4}(k+2)} &= \frac{1}{144}\left( -36 -108\zeta(2) -396\zeta(3) -666\zeta(4) -9720\zeta(5) \right. \nonumber \\ &\left.-1944\zeta(2)\zeta(3) +13245\zeta(6) +1080\zeta(3)^2 -1962\zeta(7) -720\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+2664\zeta(3)\zeta(4) -12415\zeta(8) -3312\zeta(2)\zeta(3)^2 +13824\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+3024 M(2,6)\right) \tag{910} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{5}(k+2)} &= \frac{1}{24}\left( -12 -36\zeta(2) -132\zeta(3) -222\zeta(4) +360\zeta(5) \right. \nonumber \\ &\left.+72\zeta(2)\zeta(3) +327\zeta(7) +120\zeta(2)\zeta(5) -444\zeta(3)\zeta(4) +348\zeta(9) \right. \nonumber \\ &\left.-198\zeta(3)\zeta(6) -444\zeta(4)\zeta(5) +168\zeta(2)\zeta(7) +64\zeta(3)^3\right) \tag{911} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{6}(k+2)} &= \frac{1}{5760}\left( -5760 -17280\zeta(2) -63360\zeta(3) -106560\zeta(4) +206160\zeta(6) \right. \nonumber \\ &\left.+17280\zeta(3)^2 +78480\zeta(7) +28800\zeta(2)\zeta(5) -106560\zeta(3)\zeta(4) -496600\zeta(8) \right. \nonumber \\ &\left.-132480\zeta(2)\zeta(3)^2 +552960\zeta(3)\zeta(5) +120960 M(2,6) +83520\zeta(9) -47520\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-106560\zeta(4)\zeta(5) +40320\zeta(2)\zeta(7) +15360\zeta(3)^3 -437823\zeta(10) \right. \nonumber \\ &\left.+378720\zeta(3)\zeta(7) -2520\zeta(3)^2\zeta(4) -114480\zeta(2)\zeta(3)\zeta(5) +119880\zeta(5)^2 \right. \nonumber \\ &\left.+68760 M(2,8) +11520\zeta(2) M(2,6)\right) \tag{912} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{5}(k+2)^{2}} &= \frac{1}{2304}\left( 540 +1116\zeta(2) +3276\zeta(3) +3474\zeta(4) +3240\zeta(5) \right. \nonumber \\ &\left.+792\zeta(2)\zeta(3) -14325\zeta(6) -1080\zeta(3)^2 +9990\zeta(7) +2160\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-9288\zeta(3)\zeta(4) +59332\zeta(8) +16128\zeta(2)\zeta(3)^2 -66816\zeta(3)\zeta(5) -14976 M(2,6) \right. \nonumber \\ &\left.+20928\zeta(9) -13392\zeta(3)\zeta(6) -12384\zeta(4)\zeta(5) +4032\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+1920\zeta(3)^3\right) \tag{913} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{4}(k+1)(k+2)^{2}} &= \frac{-1}{576}\left( -288 -612\zeta(2) -1836\zeta(3) -2070\zeta(4) +15120\zeta(5) \right. \nonumber \\ &\left.+2952\zeta(2)\zeta(3) -14865\zeta(6) -1080\zeta(3)^2 +6660\zeta(7) +1440\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-6192\zeta(3)\zeta(4) +14833\zeta(8) +4032\zeta(2)\zeta(3)^2 -16704\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-3744 M(2,6)\right) \tag{914} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+1)^{2}(k+2)^{2}} &= \frac{-1}{64}\left( -68 -148\zeta(2) -452\zeta(3) -534\zeta(4) -2520\zeta(5) \right. \nonumber \\ &\left.-520\zeta(2)\zeta(3) +4535\zeta(6) +360\zeta(3)^2 -370\zeta(7) -80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+344\zeta(3)\zeta(4)\right) \tag{915} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)^{3}(k+2)^{2}} &= \frac{-1}{8}\left( -18 -40\zeta(2) -124\zeta(3) -152\zeta(4) +300\zeta(5) \right. \nonumber \\ &\left.+56\zeta(2)\zeta(3) -10\zeta(6) +109\zeta(7) +40\zeta(2)\zeta(5) -148\zeta(3)\zeta(4)\right) \tag{916} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{4}(k+2)^{2}} &= \frac{1}{144}\left( 684 +1548\zeta(2) +4860\zeta(3) +6138\zeta(4) -1080\zeta(5) \right. \nonumber \\ &\left.-72\zeta(2)\zeta(3) -12885\zeta(6) -1080\zeta(3)^2 -1962\zeta(7) -720\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+2664\zeta(3)\zeta(4) +12415\zeta(8) +3312\zeta(2)\zeta(3)^2 -13824\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-3024 M(2,6)\right) \tag{917} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{5}(k+2)^{2}} &= \frac{-1}{72}\left( -720 -1656\zeta(2) -5256\zeta(3) -6804\zeta(4) +2160\zeta(5) \right. \nonumber \\ &\left.+288\zeta(2)\zeta(3) +12885\zeta(6) +1080\zeta(3)^2 +2943\zeta(7) +1080\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-3996\zeta(3)\zeta(4) -12415\zeta(8) -3312\zeta(2)\zeta(3)^2 +13824\zeta(3)\zeta(5) +3024 M(2,6) \right. \nonumber \\ &\left.+1044\zeta(9) -594\zeta(3)\zeta(6) -1332\zeta(4)\zeta(5) +504\zeta(2)\zeta(7) +192\zeta(3)^3\right) \tag{918} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{4}(k+2)^{3}} &= \frac{-1}{1152}\left( 1980 +2700\zeta(2) +6012\zeta(3) +2502\zeta(4) -432\zeta(5) \right. \nonumber \\ &\left.-216\zeta(2)\zeta(3) -13371\zeta(6) -1656\zeta(3)^2 +5976\zeta(7) +1440\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-5976\zeta(3)\zeta(4) +14833\zeta(8) +4032\zeta(2)\zeta(3)^2 -16704\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-3744 M(2,6)\right) \tag{919} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+1)(k+2)^{3}} &= \frac{-1}{32}\left( 126 +184\zeta(2) +436\zeta(3) +254\zeta(4) -864\zeta(5) \right. \nonumber \\ &\left.-176\zeta(2)\zeta(3) +83\zeta(6) -32\zeta(3)^2 -38\zeta(7) +12\zeta(3)\zeta(4)\right) \tag{920} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)^{2}(k+2)^{3}} &= \frac{1}{64}\left( -572 -884\zeta(2) -2196\zeta(3) -1550\zeta(4) +936\zeta(5) \right. \nonumber \\ &\left.+184\zeta(2)\zeta(3) +4203\zeta(6) +488\zeta(3)^2 -218\zeta(7) -80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+296\zeta(3)\zeta(4)\right) \tag{921} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{3}(k+2)^{3}} &= \frac{-1}{32}\left( 644 +1044\zeta(2) +2692\zeta(3) +2158\zeta(4) -2136\zeta(5) \right. \nonumber \\ &\left.-408\zeta(2)\zeta(3) -4163\zeta(6) -488\zeta(3)^2 -218\zeta(7) -80\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+296\zeta(3)\zeta(4)\right) \tag{922} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{4}(k+2)^{3}} &= \frac{1}{144}\left( -6480 -10944\zeta(2) -29088\zeta(3) -25560\zeta(4) +20304\zeta(5) \right. \nonumber \\ &\left.+3744\zeta(2)\zeta(3) +50352\zeta(6) +5472\zeta(3)^2 +3924\zeta(7) +1440\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-5328\zeta(3)\zeta(4) -12415\zeta(8) -3312\zeta(2)\zeta(3)^2 +13824\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+3024 M(2,6)\right) \tag{923} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+2)^{4}} &= \frac{1}{1152}\left( 9540 +8172\zeta(2) +12492\zeta(3) -4626\zeta(4) -8496\zeta(5) \right. \nonumber \\ &\left.-3672\zeta(2)\zeta(3) -11967\zeta(6) -3096\zeta(3)^2 +324\zeta(7) +288\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-792\zeta(3)\zeta(4) +12415\zeta(8) +3312\zeta(2)\zeta(3)^2 -13824\zeta(3)\zeta(5) -3024 M(2,6)\right) \tag{924} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)(k+2)^{4}} &= \frac{1}{576}\left( 11808 +11484\zeta(2) +20340\zeta(3) -54\zeta(4) -24048\zeta(5) \right. \nonumber \\ &\left.-6840\zeta(2)\zeta(3) -10473\zeta(6) -3672\zeta(3)^2 -360\zeta(7) +288\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-576\zeta(3)\zeta(4) +12415\zeta(8) +3312\zeta(2)\zeta(3)^2 -13824\zeta(3)\zeta(5) -3024 M(2,6)\right) \tag{925} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{2}(k+2)^{4}} &= \frac{1}{576}\left( 28764 +30924\zeta(2) +60444\zeta(3) +13842\zeta(4) -56520\zeta(5) \right. \nonumber \\ &\left.-15336\zeta(2)\zeta(3) -58773\zeta(6) -11736\zeta(3)^2 +1242\zeta(7) +1296\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-3816\zeta(3)\zeta(4) +24830\zeta(8) +6624\zeta(2)\zeta(3)^2 -27648\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-6048 M(2,6)\right) \tag{926} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{3}(k+2)^{4}} &= \frac{-1}{144}\left( -17280 -20160\zeta(2) -42336\zeta(3) -16632\zeta(4) +37872\zeta(5) \right. \nonumber \\ &\left.+9504\zeta(2)\zeta(3) +48120\zeta(6) +8064\zeta(3)^2 +360\zeta(7) -288\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+576\zeta(3)\zeta(4) -12415\zeta(8) -3312\zeta(2)\zeta(3)^2 +13824\zeta(3)\zeta(5) +3024 M(2,6)\right) \tag{927} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+2)^{5}} &= \frac{1}{2304}\left( -68580 -34812\zeta(2) -27900\zeta(3) +56070\zeta(4) +39960\zeta(5) \right. \nonumber \\ &\left.+17064\zeta(2)\zeta(3) +17889\zeta(6) +2232\zeta(3)^2 +24930\zeta(7) +10512\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-31752\zeta(3)\zeta(4) -50692\zeta(8) -16128\zeta(2)\zeta(3)^2 +62208\zeta(3)\zeta(5) +11520 M(2,6) \right. \nonumber \\ &\left.-8352\zeta(9) +4752\zeta(3)\zeta(6) +10656\zeta(4)\zeta(5) -4032\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-1536\zeta(3)^3\right) \tag{928} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)(k+2)^{5}} &= \frac{1}{384}\left( -30732 -19260\zeta(2) -22860\zeta(3) +18726\zeta(4) +29352\zeta(5) \right. \nonumber \\ &\left.+10248\zeta(2)\zeta(3) +12945\zeta(6) +3192\zeta(3)^2 +8550\zeta(7) +3312\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-10200\zeta(3)\zeta(4) -25174\zeta(8) -7584\zeta(2)\zeta(3)^2 +29952\zeta(3)\zeta(5) +5856 M(2,6) \right. \nonumber \\ &\left.-2784\zeta(9) +1584\zeta(3)\zeta(6) +3552\zeta(4)\zeta(5) -1344\zeta(2)\zeta(7) -512\zeta(3)^3\right) \tag{929} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{2}(k+2)^{5}} &= \frac{1}{72}\left( -15120 -11088\zeta(2) -16128\zeta(3) +5292\zeta(4) +18072\zeta(5) \right. \nonumber \\ &\left.+5760\zeta(2)\zeta(3) +12201\zeta(6) +2664\zeta(3)^2 +3051\zeta(7) +1080\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-3348\zeta(3)\zeta(4) -12544\zeta(8) -3672\zeta(2)\zeta(3)^2 +14688\zeta(3)\zeta(5) +2952 M(2,6) \right. \nonumber \\ &\left.-1044\zeta(9) +594\zeta(3)\zeta(6) +1332\zeta(4)\zeta(5) -504\zeta(2)\zeta(7) -192\zeta(3)^3\right) \tag{930} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+2)^{6}} &= \frac{-1}{11520}\left( -990540 -275580\zeta(2) +20340\zeta(3) +706950\zeta(4) +394920\zeta(5) \right. \nonumber \\ &\left.+76680\zeta(2)\zeta(3) +216465\zeta(6) -99720\zeta(3)^2 +272790\zeta(7) +140400\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-385560\zeta(3)\zeta(4) -260590\zeta(8) -76320\zeta(2)\zeta(3)^2 +311040\zeta(3)\zeta(5) +56160 M(2,6) \right. \nonumber \\ &\left.+147360\zeta(9) -82800\zeta(3)\zeta(6) -136800\zeta(4)\zeta(5) +48960\zeta(2)\zeta(7) +15360\zeta(3)^3 \right. \nonumber \\ &\left.-437823\zeta(10) +378720\zeta(3)\zeta(7) -2520\zeta(3)^2\zeta(4) -114480\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+119880\zeta(5)^2 +68760 M(2,8) +11520\zeta(2) M(2,6)\right) \tag{931} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)(k+2)^{6}} &= \frac{-1}{5760}\left( -1451520 -564480\zeta(2) -322560\zeta(3) +987840\zeta(4) \right. \nonumber \\ &\left.+835200\zeta(5) +230400\zeta(2)\zeta(3) +410640\zeta(6) -51840\zeta(3)^2 +401040\zeta(7) \right. \nonumber \\ &\left.+190080\zeta(2)\zeta(5) -538560\zeta(3)\zeta(4) -638200\zeta(8) -190080\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+760320\zeta(3)\zeta(5) +144000 M(2,6) +105600\zeta(9) -59040\zeta(3)\zeta(6) -83520\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+28800\zeta(2)\zeta(7) +7680\zeta(3)^3 -437823\zeta(10) +378720\zeta(3)\zeta(7) -2520\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.-114480\zeta(2)\zeta(3)\zeta(5) +119880\zeta(5)^2 +68760 M(2,8) +11520\zeta(2) M(2,6)\right) \tag{932} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+2)^{7}} &= \frac{-1}{240}\left( 50400 +6720\zeta(2) -10080\zeta(3) -27720\zeta(4) -18000\zeta(5) \right. \nonumber \\ &\left.+1440\zeta(2)\zeta(3) -12180\zeta(6) +6720\zeta(3)^2 -11700\zeta(7) -3360\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+12960\zeta(3)\zeta(4) +9310\zeta(8) +1200\zeta(2)\zeta(3)^2 -7680\zeta(3)\zeta(5) -2640 M(2,6) \right. \nonumber \\ &\left.-8120\zeta(9) +6600\zeta(3)\zeta(6) +8640\zeta(4)\zeta(5) -4320\zeta(2)\zeta(7) -1440\zeta(3)^3 \right. \nonumber \\ &\left.+3006\zeta(10) -4800\zeta(3)\zeta(7) -480\zeta(3)^2\zeta(4) +3360\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-2880\zeta(5)^2 -240 M(2,8) +1185\zeta(11) +1840\zeta(2)\zeta(9) -430\zeta(3)\zeta(8) \right. \nonumber \\ &\left.-3420\zeta(4)\zeta(7) -1010\zeta(5)\zeta(6) -400\zeta(2)\zeta(3)^3 +1440\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.+80 M(3,8)\right) \tag{933} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{6}} &= \frac{-1}{576}\left( 781671\zeta(11) +88016\zeta(2)\zeta(9) -296660\zeta(3)\zeta(8) \right. \nonumber \\ &\left.-411984\zeta(4)\zeta(7) -220080\zeta(5)\zeta(6) -21120\zeta(2)\zeta(3)^3 +141120\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.-8640\zeta(3) M(2,6) -27840 M(3,8)\right) \tag{934} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{5}(k+1)} &= \frac{-1}{2304}\left( -411264\zeta(6) -51840\zeta(3)^2 +295344\zeta(7) +65664\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+76032\zeta(3)\zeta(4) +542488\zeta(8) +152640\zeta(2)\zeta(3)^2 -630144\zeta(3)\zeta(5) -135360 M(2,6) \right. \nonumber \\ &\left.+302144\zeta(9) -469920\zeta(3)\zeta(6) +152064\zeta(4)\zeta(5) +76320\zeta(2)\zeta(7) -11520\zeta(3)^3 \right. \nonumber \\ &\left.+579897\zeta(10) -519840\zeta(3)\zeta(7) +3240\zeta(3)^2\zeta(4) +185040\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-203832\zeta(5)^2 -98280 M(2,8) -11520\zeta(2) M(2,6)\right) \tag{935} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{4}(k+1)^{2}} &= \frac{1}{144}\left( -102816\zeta(6) -12960\zeta(3)^2 +72072\zeta(7) +16416\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+19008\zeta(3)\zeta(4) +67811\zeta(8) +19080\zeta(2)\zeta(3)^2 -78768\zeta(3)\zeta(5) -16920 M(2,6) \right. \nonumber \\ &\left.+18884\zeta(9) -29370\zeta(3)\zeta(6) +9504\zeta(4)\zeta(5) +4770\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-720\zeta(3)^3\right) \tag{936} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{3}(k+1)^{3}} &= \frac{1}{72}\left( 77112\zeta(6) +9720\zeta(3)^2 -52731\zeta(7) -12312\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-14256\zeta(3)\zeta(4) -33358\zeta(8) -9180\zeta(2)\zeta(3)^2 +37800\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+8100 M(2,6)\right) \tag{937} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+1)^{4}} &= \frac{-1}{144}\left( 102816\zeta(6) +12960\zeta(3)^2 -68544\zeta(7) -16416\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-19008\zeta(3)\zeta(4) -65621\zeta(8) -17640\zeta(2)\zeta(3)^2 +72432\zeta(3)\zeta(5) +15480 M(2,6) \right. \nonumber \\ &\left.-14240\zeta(9) +25770\zeta(3)\zeta(6) -9504\zeta(4)\zeta(5) -4770\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+720\zeta(3)^3\right) \tag{938} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)^{5}} &= \frac{-1}{2304}\left( -411264\zeta(6) -51840\zeta(3)^2 +267120\zeta(7) +65664\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+76032\zeta(3)\zeta(4) +524968\zeta(8) +141120\zeta(2)\zeta(3)^2 -579456\zeta(3)\zeta(5) -123840 M(2,6) \right. \nonumber \\ &\left.+227840\zeta(9) -412320\zeta(3)\zeta(6) +152064\zeta(4)\zeta(5) +76320\zeta(2)\zeta(7) -11520\zeta(3)^3 \right. \nonumber \\ &\left.+449109\zeta(10) -387360\zeta(3)\zeta(7) -9720\zeta(3)^2\zeta(4) +124560\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-122328\zeta(5)^2 -68040 M(2,8) -11520\zeta(2) M(2,6)\right) \tag{939} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{6}} &= \frac{-1}{576}\left( 667227\zeta(11) +68816\zeta(2)\zeta(9) -248300\zeta(3)\zeta(8) \right. \nonumber \\ &\left.-350784\zeta(4)\zeta(7) -176280\zeta(5)\zeta(6) -16320\zeta(2)\zeta(3)^3 +112320\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.-8640\zeta(3) M(2,6) -23040 M(3,8)\right) \tag{940} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{5}(k+2)} &= \frac{-1}{4608}\left( -144 -576\zeta(2) -3024\zeta(3) -9036\zeta(4) -10224\zeta(5) \right. \nonumber \\ &\left.-2160\zeta(2)\zeta(3) -25704\zeta(6) -3240\zeta(3)^2 +36918\zeta(7) +8208\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+9504\zeta(3)\zeta(4) +135622\zeta(8) +38160\zeta(2)\zeta(3)^2 -157536\zeta(3)\zeta(5) -33840 M(2,6) \right. \nonumber \\ &\left.+151072\zeta(9) -234960\zeta(3)\zeta(6) +76032\zeta(4)\zeta(5) +38160\zeta(2)\zeta(7) -5760\zeta(3)^3 \right. \nonumber \\ &\left.+579897\zeta(10) -519840\zeta(3)\zeta(7) +3240\zeta(3)^2\zeta(4) +185040\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-203832\zeta(5)^2 -98280 M(2,8) -11520\zeta(2) M(2,6)\right) \tag{941} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{4}(k+1)(k+2)} &= \frac{1}{1152}\left( 72 +288\zeta(2) +1512\zeta(3) +4518\zeta(4) +5112\zeta(5) \right. \nonumber \\ &\left.+1080\zeta(2)\zeta(3) -192780\zeta(6) -24300\zeta(3)^2 +129213\zeta(7) +28728\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+33264\zeta(3)\zeta(4) +203433\zeta(8) +57240\zeta(2)\zeta(3)^2 -236304\zeta(3)\zeta(5) -50760 M(2,6) \right. \nonumber \\ &\left.+75536\zeta(9) -117480\zeta(3)\zeta(6) +38016\zeta(4)\zeta(5) +19080\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-2880\zeta(3)^3\right) \tag{942} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{3}(k+1)^{2}(k+2)} &= \frac{1}{576}\left( 72 +288\zeta(2) +1512\zeta(3) +4518\zeta(4) +5112\zeta(5) \right. \nonumber \\ &\left.+1080\zeta(2)\zeta(3) +218484\zeta(6) +27540\zeta(3)^2 -159075\zeta(7) -36936\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-42768\zeta(3)\zeta(4) -67811\zeta(8) -19080\zeta(2)\zeta(3)^2 +78768\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+16920 M(2,6)\right) \tag{943} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+1)^{3}(k+2)} &= \frac{-1}{288}\left( -72 -288\zeta(2) -1512\zeta(3) -4518\zeta(4) -5112\zeta(5) \right. \nonumber \\ &\left.-1080\zeta(2)\zeta(3) +89964\zeta(6) +11340\zeta(3)^2 -51849\zeta(7) -12312\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-14256\zeta(3)\zeta(4) -65621\zeta(8) -17640\zeta(2)\zeta(3)^2 +72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+15480 M(2,6)\right) \tag{944} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)^{4}(k+2)} &= \frac{-1}{144}\left( -72 -288\zeta(2) -1512\zeta(3) -4518\zeta(4) -5112\zeta(5) \right. \nonumber \\ &\left.-1080\zeta(2)\zeta(3) -12852\zeta(6) -1620\zeta(3)^2 +16695\zeta(7) +4104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+4752\zeta(3)\zeta(4) +14240\zeta(9) -25770\zeta(3)\zeta(6) +9504\zeta(4)\zeta(5) +4770\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-720\zeta(3)^3\right) \tag{945} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{5}(k+2)} &= \frac{1}{2304}\left( 2304 +9216\zeta(2) +48384\zeta(3) +144576\zeta(4) +163584\zeta(5) \right. \nonumber \\ &\left.+34560\zeta(2)\zeta(3) -267120\zeta(7) -65664\zeta(2)\zeta(5) -76032\zeta(3)\zeta(4) +524968\zeta(8) \right. \nonumber \\ &\left.+141120\zeta(2)\zeta(3)^2 -579456\zeta(3)\zeta(5) -123840 M(2,6) -227840\zeta(9) \right. \nonumber \\ &\left.+412320\zeta(3)\zeta(6) -152064\zeta(4)\zeta(5) -76320\zeta(2)\zeta(7) +11520\zeta(3)^3 \right. \nonumber \\ &\left.+449109\zeta(10) -387360\zeta(3)\zeta(7) -9720\zeta(3)^2\zeta(4) +124560\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-122328\zeta(5)^2 -68040 M(2,8) -11520\zeta(2) M(2,6)\right) \tag{946} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{4}(k+2)^{2}} &= \frac{1}{1152}\left( -576 -1656\zeta(2) -7200\zeta(3) -16092\zeta(4) -9936\zeta(5) \right. \nonumber \\ &\left.-2520\zeta(2)\zeta(3) -12819\zeta(6) -2160\zeta(3)^2 +36036\zeta(7) +8208\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+9504\zeta(3)\zeta(4) +67811\zeta(8) +19080\zeta(2)\zeta(3)^2 -78768\zeta(3)\zeta(5) -16920 M(2,6) \right. \nonumber \\ &\left.+37768\zeta(9) -58740\zeta(3)\zeta(6) +19008\zeta(4)\zeta(5) +9540\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-1440\zeta(3)^3\right) \tag{947} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{3}(k+1)(k+2)^{2}} &= \frac{-1}{1152}\left( 1224 +3600\zeta(2) +15912\zeta(3) +36702\zeta(4) +24984\zeta(5) \right. \nonumber \\ &\left.+6120\zeta(2)\zeta(3) -167142\zeta(6) -19980\zeta(3)^2 +57141\zeta(7) +12312\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+14256\zeta(3)\zeta(4) +67811\zeta(8) +19080\zeta(2)\zeta(3)^2 -78768\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-16920 M(2,6)\right) \tag{948} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+1)^{2}(k+2)^{2}} &= \frac{-1}{96}\left( 216 +648\zeta(2) +2904\zeta(3) +6870\zeta(4) +5016\zeta(5) \right. \nonumber \\ &\left.+1200\zeta(2)\zeta(3) +8557\zeta(6) +1260\zeta(3)^2 -16989\zeta(7) -4104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-4752\zeta(3)\zeta(4)\right) \tag{949} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)^{3}(k+2)^{2}} &= \frac{-1}{288}\left( 1368 +4176\zeta(2) +18936\zeta(3) +45738\zeta(4) +35208\zeta(5) \right. \nonumber \\ &\left.+8280\zeta(2)\zeta(3) -38622\zeta(6) -3780\zeta(3)^2 -50085\zeta(7) -12312\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-14256\zeta(3)\zeta(4) +65621\zeta(8) +17640\zeta(2)\zeta(3)^2 -72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-15480 M(2,6)\right) \tag{950} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{4}(k+2)^{2}} &= \frac{1}{144}\left( -1440 -4464\zeta(2) -20448\zeta(3) -50256\zeta(4) -40320\zeta(5) \right. \nonumber \\ &\left.-9360\zeta(2)\zeta(3) +25770\zeta(6) +2160\zeta(3)^2 +66780\zeta(7) +16416\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+19008\zeta(3)\zeta(4) -65621\zeta(8) -17640\zeta(2)\zeta(3)^2 +72432\zeta(3)\zeta(5) +15480 M(2,6) \right. \nonumber \\ &\left.+14240\zeta(9) -25770\zeta(3)\zeta(6) +9504\zeta(4)\zeta(5) +4770\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-720\zeta(3)^3\right) \tag{951} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{3}(k+2)^{3}} &= \frac{-1}{1152}\left( -4536 -9144\zeta(2) -32184\zeta(3) -49770\zeta(4) -5256\zeta(5) \right. \nonumber \\ &\left.-2160\zeta(2)\zeta(3) +22899\zeta(6) +3420\zeta(3)^2 +42921\zeta(7) +8712\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+27576\zeta(3)\zeta(4) +66716\zeta(8) +18360\zeta(2)\zeta(3)^2 -75600\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-16200 M(2,6)\right) \tag{952} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+1)(k+2)^{3}} &= \frac{1}{1152}\left( 10296 +21888\zeta(2) +80280\zeta(3) +136242\zeta(4) \right. \nonumber \\ &\left.+35496\zeta(5) +10440\zeta(2)\zeta(3) -212940\zeta(6) -26820\zeta(3)^2 -28701\zeta(7) \right. \nonumber \\ &\left.-5112\zeta(2)\zeta(5) -40896\zeta(3)\zeta(4) -65621\zeta(8) -17640\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+72432\zeta(3)\zeta(5) +15480 M(2,6)\right) \tag{953} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)^{2}(k+2)^{3}} &= \frac{1}{576}\left( 11592 +25776\zeta(2) +97704\zeta(3) +177462\zeta(4) +65592\zeta(5) \right. \nonumber \\ &\left.+17640\zeta(2)\zeta(3) -161598\zeta(6) -19260\zeta(3)^2 -130635\zeta(7) -29736\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-69408\zeta(3)\zeta(4) -65621\zeta(8) -17640\zeta(2)\zeta(3)^2 +72432\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+15480 M(2,6)\right) \tag{954} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{3}(k+2)^{3}} &= \frac{-1}{24}\left( -1080 -2496\zeta(2) -9720\zeta(3) -18600\zeta(4) -8400\zeta(5) \right. \nonumber \\ &\left.-2160\zeta(2)\zeta(3) +16685\zeta(6) +1920\zeta(3)^2 +15060\zeta(7) +3504\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+6972\zeta(3)\zeta(4)\right) \tag{955} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+2)^{4}} &= \frac{-1}{1152}\left( 23616 +32616\zeta(2) +90432\zeta(3) +83340\zeta(4) -43344\zeta(5) \right. \nonumber \\ &\left.-12600\zeta(2)\zeta(3) -102651\zeta(6) -23040\zeta(3)^2 -16452\zeta(7) +432\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-39024\zeta(3)\zeta(4) +58529\zeta(8) +15480\zeta(2)\zeta(3)^2 -65808\zeta(3)\zeta(5) -14760 M(2,6) \right. \nonumber \\ &\left.-28480\zeta(9) +51540\zeta(3)\zeta(6) -19008\zeta(4)\zeta(5) -9540\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+1440\zeta(3)^3\right) \tag{956} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)(k+2)^{4}} &= \frac{1}{1152}\left( -57528 -87120\zeta(2) -261144\zeta(3) -302922\zeta(4) +51192\zeta(5) \right. \nonumber \\ &\left.+14760\zeta(2)\zeta(3) +418242\zeta(6) +72900\zeta(3)^2 +61605\zeta(7) +4248\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+118944\zeta(3)\zeta(4) -51437\zeta(8) -13320\zeta(2)\zeta(3)^2 +59184\zeta(3)\zeta(5) +14040 M(2,6) \right. \nonumber \\ &\left.+56960\zeta(9) -103080\zeta(3)\zeta(6) +38016\zeta(4)\zeta(5) +19080\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-2880\zeta(3)^3\right) \tag{957} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)^{2}(k+2)^{4}} &= \frac{1}{72}\left( -8640 -14112\zeta(2) -44856\zeta(3) -60048\zeta(4) -1800\zeta(5) \right. \nonumber \\ &\left.-360\zeta(2)\zeta(3) +72480\zeta(6) +11520\zeta(3)^2 +24030\zeta(7) +4248\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+23544\zeta(3)\zeta(4) +1773\zeta(8) +540\zeta(2)\zeta(3)^2 -1656\zeta(3)\zeta(5) -180 M(2,6) \right. \nonumber \\ &\left.+7120\zeta(9) -12885\zeta(3)\zeta(6) +4752\zeta(4)\zeta(5) +2385\zeta(2)\zeta(7) -360\zeta(3)^3\right) \tag{958} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+2)^{5}} &= \frac{-1}{4608}\left( -368784 -341856\zeta(2) -719568\zeta(3) -238572\zeta(4) +579600\zeta(5) \right. \nonumber \\ &\left.+200880\zeta(2)\zeta(3) +678396\zeta(6) +171000\zeta(3)^2 +164070\zeta(7) +45648\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-104256\zeta(3)\zeta(4) -1368878\zeta(8) -390960\zeta(2)\zeta(3)^2 +1583136\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+326160 M(2,6) -53120\zeta(9) -111120\zeta(3)\zeta(6) +289152\zeta(4)\zeta(5) -42480\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-36480\zeta(3)^3 +449109\zeta(10) -387360\zeta(3)\zeta(7) -9720\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+124560\zeta(2)\zeta(3)\zeta(5) -122328\zeta(5)^2 -68040 M(2,8) -11520\zeta(2) M(2,6)\right) \tag{959} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+1)(k+2)^{5}} &= \frac{-1}{2304}\left( -483840 -516096\zeta(2) -1241856\zeta(3) -844416\zeta(4) \right. \nonumber \\ &\left.+681984\zeta(5) +230400\zeta(2)\zeta(3) +1514880\zeta(6) +316800\zeta(3)^2 +287280\zeta(7) \right. \nonumber \\ &\left.+54144\zeta(2)\zeta(5) +133632\zeta(3)\zeta(4) -1471752\zeta(8) -417600\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+1701504\zeta(3)\zeta(5) +354240 M(2,6) +60800\zeta(9) -317280\zeta(3)\zeta(6) +365184\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-4320\zeta(2)\zeta(7) -42240\zeta(3)^3 +449109\zeta(10) -387360\zeta(3)\zeta(7) -9720\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+124560\zeta(2)\zeta(3)\zeta(5) -122328\zeta(5)^2 -68040 M(2,8) -11520\zeta(2) M(2,6)\right) \tag{960} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{(k+2)^{6}} &= \frac{-1}{1152}\left( 290304 +177408\zeta(2) +266112\zeta(3) -87552\zeta(4) -298368\zeta(5) \right. \nonumber \\ &\left.-97920\zeta(2)\zeta(3) -236112\zeta(6) -28800\zeta(3)^2 -161280\zeta(7) -69120\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+201600\zeta(3)\zeta(4) +547240\zeta(8) +161280\zeta(2)\zeta(3)^2 -645120\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-126720 M(2,6) -11040\zeta(9) +5760\zeta(3)\zeta(6) -11520\zeta(4)\zeta(5) +5760\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+3840\zeta(3)^3 +437823\zeta(10) -378720\zeta(3)\zeta(7) +2520\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+114480\zeta(2)\zeta(3)\zeta(5) -119880\zeta(5)^2 -68760 M(2,8) -11520\zeta(2) M(2,6) +1334454\zeta(11) \right. \nonumber \\ &\left.+137632\zeta(2)\zeta(9) -496600\zeta(3)\zeta(8) -701568\zeta(4)\zeta(7) -352560\zeta(5)\zeta(6) \right. \nonumber \\ &\left.-32640\zeta(2)\zeta(3)^3 +224640\zeta(3)^2\zeta(5) -17280\zeta(3) M(2,6) -46080 M(3,8)\right) \tag{961} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{5}} &= \frac{1}{192}\left( -734643\zeta(11) -83472\zeta(2)\zeta(9) +271244\zeta(3)\zeta(8) \right. \nonumber \\ &\left.+395088\zeta(4)\zeta(7) +205424\zeta(5)\zeta(6) +19360\zeta(2)\zeta(3)^3 -130176\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.+9120\zeta(3) M(2,6) +25600 M(3,8)\right) \tag{962} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{4}(k+1)} &= \frac{-1}{384}\left( 247296\zeta(7) +55680\zeta(2)\zeta(5) +114048\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-280464\zeta(8) +15744\zeta(2)\zeta(3)^2 -187008\zeta(3)\zeta(5) -21888 M(2,6) +119584\zeta(9) \right. \nonumber \\ &\left.-209952\zeta(3)\zeta(6) +96768\zeta(4)\zeta(5) +31248\zeta(2)\zeta(7) -8704\zeta(3)^3 +814101\zeta(10) \right. \nonumber \\ &\left.-529680\zeta(3)\zeta(7) +253944\zeta(3)^2\zeta(4) +1200\zeta(2)\zeta(3)\zeta(5) -365064\zeta(5)^2 \right. \nonumber \\ &\left.-103128 M(2,8) -45120\zeta(2) M(2,6)\right) \tag{963} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{3}(k+1)^{2}} &= \frac{1}{24}\left( 46368\zeta(7) +10440\zeta(2)\zeta(5) +21384\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-52085\zeta(8) +2892\zeta(2)\zeta(3)^2 -34704\zeta(3)\zeta(5) -4044 M(2,6) +7474\zeta(9) \right. \nonumber \\ &\left.-13122\zeta(3)\zeta(6) +6048\zeta(4)\zeta(5) +1953\zeta(2)\zeta(7) -544\zeta(3)^3\right) \tag{964} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{2}(k+1)^{3}} &= \frac{1}{24}\left( -46368\zeta(7) -10440\zeta(2)\zeta(5) -21384\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+51583\zeta(8) -2832\zeta(2)\zeta(3)^2 +34344\zeta(3)\zeta(5) +3984 M(2,6) -6146\zeta(9) \right. \nonumber \\ &\left.+12582\zeta(3)\zeta(6) -5832\zeta(4)\zeta(5) -1953\zeta(2)\zeta(7) +536\zeta(3)^3\right) \tag{965} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+1)^{4}} &= \frac{-1}{384}\left( -247296\zeta(7) -55680\zeta(2)\zeta(5) -114048\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+272432\zeta(8) -14784\zeta(2)\zeta(3)^2 +181248\zeta(3)\zeta(5) +20928 M(2,6) -98336\zeta(9) \right. \nonumber \\ &\left.+201312\zeta(3)\zeta(6) -93312\zeta(4)\zeta(5) -31248\zeta(2)\zeta(7) +8576\zeta(3)^3 -779835\zeta(10) \right. \nonumber \\ &\left.+490704\zeta(3)\zeta(7) -245544\zeta(3)^2\zeta(4) +15600\zeta(2)\zeta(3)\zeta(5) +339864\zeta(5)^2 \right. \nonumber \\ &\left.+94728 M(2,8) +45120\zeta(2) M(2,6)\right) \tag{966} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)^{5}} &= \frac{1}{192}\left( 686799\zeta(11) +74512\zeta(2)\zeta(9) -262484\zeta(3)\zeta(8) \right. \nonumber \\ &\left.-362208\zeta(4)\zeta(7) -182584\zeta(5)\zeta(6) -18080\zeta(2)\zeta(3)^3 +120384\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.-8160\zeta(3) M(2,6) -23360 M(3,8)\right) \tag{967} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{4}(k+2)} &= \frac{1}{768}\left( -48 -240\zeta(2) -1632\zeta(3) -6852\zeta(4) -13704\zeta(5) \right. \nonumber \\ &\left.-2928\zeta(2)\zeta(3) -25164\zeta(6) -3216\zeta(3)^2 -30912\zeta(7) -6960\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-14256\zeta(3)\zeta(4) +70116\zeta(8) -3936\zeta(2)\zeta(3)^2 +46752\zeta(3)\zeta(5) +5472 M(2,6) \right. \nonumber \\ &\left.-59792\zeta(9) +104976\zeta(3)\zeta(6) -48384\zeta(4)\zeta(5) -15624\zeta(2)\zeta(7) +4352\zeta(3)^3 \right. \nonumber \\ &\left.-814101\zeta(10) +529680\zeta(3)\zeta(7) -253944\zeta(3)^2\zeta(4) -1200\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+365064\zeta(5)^2 +103128 M(2,8) +45120\zeta(2) M(2,6)\right) \tag{968} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{3}(k+1)(k+2)} &= \frac{1}{96}\left( -12 -60\zeta(2) -408\zeta(3) -1713\zeta(4) -3426\zeta(5) \right. \nonumber \\ &\left.-732\zeta(2)\zeta(3) -6291\zeta(6) -804\zeta(3)^2 +54096\zeta(7) +12180\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+24948\zeta(3)\zeta(4) -52587\zeta(8) +2952\zeta(2)\zeta(3)^2 -35064\zeta(3)\zeta(5) -4104 M(2,6) \right. \nonumber \\ &\left.+14948\zeta(9) -26244\zeta(3)\zeta(6) +12096\zeta(4)\zeta(5) +3906\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-1088\zeta(3)^3\right) \tag{969} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{2}(k+1)^{2}(k+2)} &= \frac{1}{48}\left( -12 -60\zeta(2) -408\zeta(3) -1713\zeta(4) -3426\zeta(5) \right. \nonumber \\ &\left.-732\zeta(2)\zeta(3) -6291\zeta(6) -804\zeta(3)^2 -38640\zeta(7) -8700\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-17820\zeta(3)\zeta(4) +51583\zeta(8) -2832\zeta(2)\zeta(3)^2 +34344\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+3984 M(2,6)\right) \tag{970} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+1)^{3}(k+2)} &= \frac{1}{24}\left( -12 -60\zeta(2) -408\zeta(3) -1713\zeta(4) -3426\zeta(5) \right. \nonumber \\ &\left.-732\zeta(2)\zeta(3) -6291\zeta(6) -804\zeta(3)^2 +7728\zeta(7) +1740\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+3564\zeta(3)\zeta(4) +6146\zeta(9) -12582\zeta(3)\zeta(6) +5832\zeta(4)\zeta(5) +1953\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-536\zeta(3)^3\right) \tag{971} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)^{4}(k+2)} &= \frac{1}{384}\left( -384 -1920\zeta(2) -13056\zeta(3) -54816\zeta(4) -109632\zeta(5) \right. \nonumber \\ &\left.-23424\zeta(2)\zeta(3) -201312\zeta(6) -25728\zeta(3)^2 +272432\zeta(8) -14784\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+181248\zeta(3)\zeta(5) +20928 M(2,6) +98336\zeta(9) -201312\zeta(3)\zeta(6) +93312\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+31248\zeta(2)\zeta(7) -8576\zeta(3)^3 -779835\zeta(10) +490704\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-245544\zeta(3)^2\zeta(4) +15600\zeta(2)\zeta(3)\zeta(5) +339864\zeta(5)^2 +94728 M(2,8) \right. \nonumber \\ &\left.+45120\zeta(2) M(2,6)\right) \tag{972} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{3}(k+2)^{2}} &= \frac{1}{192}\left( 204 +756\zeta(2) +4344\zeta(3) +14427\zeta(4) +20382\zeta(5) \right. \nonumber \\ &\left.+4644\zeta(2)\zeta(3) +18570\zeta(6) +2940\zeta(3)^2 +6489\zeta(7) +1116\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+5940\zeta(3)\zeta(4) -52085\zeta(8) +2892\zeta(2)\zeta(3)^2 -34704\zeta(3)\zeta(5) -4044 M(2,6) \right. \nonumber \\ &\left.+14948\zeta(9) -26244\zeta(3)\zeta(6) +12096\zeta(4)\zeta(5) +3906\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-1088\zeta(3)^3\right) \tag{973} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{2}(k+1)(k+2)^{2}} &= \frac{1}{96}\left( 216 +816\zeta(2) +4752\zeta(3) +16140\zeta(4) +23808\zeta(5) \right. \nonumber \\ &\left.+5376\zeta(2)\zeta(3) +24861\zeta(6) +3744\zeta(3)^2 -47607\zeta(7) -11064\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-19008\zeta(3)\zeta(4) +502\zeta(8) -60\zeta(2)\zeta(3)^2 +360\zeta(3)\zeta(5) +60 M(2,6)\right) \tag{974} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+1)^{2}(k+2)^{2}} &= \frac{-1}{16}\left( -76 -292\zeta(2) -1720\zeta(3) -5951\zeta(4) -9078\zeta(5) \right. \nonumber \\ &\left.-2036\zeta(2)\zeta(3) -10384\zeta(6) -1516\zeta(3)^2 +2989\zeta(7) +788\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+396\zeta(3)\zeta(4) +17027\zeta(8) -924\zeta(2)\zeta(3)^2 +11328\zeta(3)\zeta(5) +1308 M(2,6)\right) \tag{975} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)^{3}(k+2)^{2}} &= \frac{1}{24}\left( 240 +936\zeta(2) +5568\zeta(3) +19566\zeta(4) +30660\zeta(5) \right. \nonumber \\ &\left.+6840\zeta(2)\zeta(3) +37443\zeta(6) +5352\zeta(3)^2 -16695\zeta(7) -4104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-4752\zeta(3)\zeta(4) -51081\zeta(8) +2772\zeta(2)\zeta(3)^2 -33984\zeta(3)\zeta(5) -3924 M(2,6) \right. \nonumber \\ &\left.-6146\zeta(9) +12582\zeta(3)\zeta(6) -5832\zeta(4)\zeta(5) -1953\zeta(2)\zeta(7) +536\zeta(3)^3\right) \tag{976} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{2}(k+2)^{3}} &= \frac{1}{192}\left( -1716 -4644\zeta(2) -22296\zeta(3) -56835\zeta(4) -51078\zeta(5) \right. \nonumber \\ &\left.-12276\zeta(2)\zeta(3) +6129\zeta(6) -444\zeta(3)^2 +33786\zeta(7) +7596\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+21636\zeta(3)\zeta(4) +117204\zeta(8) +14808\zeta(2)\zeta(3)^2 -38088\zeta(3)\zeta(5) -11496 M(2,6) \right. \nonumber \\ &\left.-12292\zeta(9) +25164\zeta(3)\zeta(6) -11664\zeta(4)\zeta(5) -3906\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+1072\zeta(3)^3\right) \tag{977} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+1)(k+2)^{3}} &= \frac{-1}{96}\left( 1932 +5460\zeta(2) +27048\zeta(3) +72975\zeta(4) +74886\zeta(5) \right. \nonumber \\ &\left.+17652\zeta(2)\zeta(3) +18732\zeta(6) +4188\zeta(3)^2 -81393\zeta(7) -18660\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-40644\zeta(3)\zeta(4) -116702\zeta(8) -14868\zeta(2)\zeta(3)^2 +38448\zeta(3)\zeta(5) +11556 M(2,6) \right. \nonumber \\ &\left.+12292\zeta(9) -25164\zeta(3)\zeta(6) +11664\zeta(4)\zeta(5) +3906\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-1072\zeta(3)^3\right) \tag{978} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)^{2}(k+2)^{3}} &= \frac{1}{48}\left( -2160 -6336\zeta(2) -32208\zeta(3) -90828\zeta(4) -102120\zeta(5) \right. \nonumber \\ &\left.-23760\zeta(2)\zeta(3) -49884\zeta(6) -8736\zeta(3)^2 +90360\zeta(7) +21024\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+41832\zeta(3)\zeta(4) +167783\zeta(8) +12096\zeta(2)\zeta(3)^2 -4464\zeta(3)\zeta(5) -7632 M(2,6) \right. \nonumber \\ &\left.-12292\zeta(9) +25164\zeta(3)\zeta(6) -11664\zeta(4)\zeta(5) -3906\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+1072\zeta(3)^3\right) \tag{979} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+2)^{4}} &= \frac{-1}{768}\left( -38352 -74928\zeta(2) -297696\zeta(3) -562980\zeta(4) -259368\zeta(5) \right. \nonumber \\ &\left.-60336\zeta(2)\zeta(3) +452160\zeta(6) +73200\zeta(3)^2 +297468\zeta(7) +51312\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+321840\zeta(3)\zeta(4) +358960\zeta(8) +75504\zeta(2)\zeta(3)^2 -270912\zeta(3)\zeta(5) -59568 M(2,6) \right. \nonumber \\ &\left.+178672\zeta(9) -311664\zeta(3)\zeta(6) +105408\zeta(4)\zeta(5) +60696\zeta(2)\zeta(7) -7232\zeta(3)^3 \right. \nonumber \\ &\left.-779835\zeta(10) +490704\zeta(3)\zeta(7) -245544\zeta(3)^2\zeta(4) +15600\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+339864\zeta(5)^2 +94728 M(2,8) +45120\zeta(2) M(2,6)\right) \tag{980} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+1)(k+2)^{4}} &= \frac{1}{384}\left( 46080 +96768\zeta(2) +405888\zeta(3) +854880\zeta(4) +558912\zeta(5) \right. \nonumber \\ &\left.+130944\zeta(2)\zeta(3) -377232\zeta(6) -56448\zeta(3)^2 -623040\zeta(7) -125952\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-484416\zeta(3)\zeta(4) -825768\zeta(8) -134976\zeta(2)\zeta(3)^2 +424704\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+105792 M(2,6) -129504\zeta(9) +211008\zeta(3)\zeta(6) -58752\zeta(4)\zeta(5) -45072\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+2944\zeta(3)^3 +779835\zeta(10) -490704\zeta(3)\zeta(7) +245544\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.-15600\zeta(2)\zeta(3)\zeta(5) -339864\zeta(5)^2 -94728 M(2,8) -45120\zeta(2) M(2,6)\right) \tag{981} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{6}}{(k+2)^{5}} &= \frac{-1}{384}\left( 80640 +112896\zeta(2) +368256\zeta(3) +494496\zeta(4) +35136\zeta(5) \right. \nonumber \\ &\left.-5760\zeta(2)\zeta(3) -565536\zeta(6) -111360\zeta(3)^2 -197280\zeta(7) -31104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-202752\zeta(3)\zeta(4) +471672\zeta(8) +133440\zeta(2)\zeta(3)^2 -549504\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-116160 M(2,6) -144320\zeta(9) +364800\zeta(3)\zeta(6) -258624\zeta(4)\zeta(5) -36000\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+26880\zeta(3)^3 -449109\zeta(10) +387360\zeta(3)\zeta(7) +9720\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.-124560\zeta(2)\zeta(3)\zeta(5) +122328\zeta(5)^2 +68040 M(2,8) +11520\zeta(2) M(2,6) -1373598\zeta(11) \right. \nonumber \\ &\left.-149024\zeta(2)\zeta(9) +524968\zeta(3)\zeta(8) +724416\zeta(4)\zeta(7) +365168\zeta(5)\zeta(6) \right. \nonumber \\ &\left.+36160\zeta(2)\zeta(3)^3 -240768\zeta(3)^2\zeta(5) +16320\zeta(3) M(2,6) +46720 M(3,8)\right) \tag{982} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k^{4}} &= \frac{1}{1152}\left( -16370805\zeta(11) -1684144\zeta(2)\zeta(9) -5889744\zeta(3)\zeta(8) \right. \nonumber \\ &\left.+10724760\zeta(4)\zeta(7) +10480104\zeta(5)\zeta(6) -844032\zeta(2)\zeta(3)^3 \right. \nonumber \\ &\left.+2330496\zeta(3)^2\zeta(5) +1431360\zeta(3) M(2,6) +630336 M(3,8)\right) \tag{983} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k^{3}(k+1)} &= \frac{-1}{23040}\left( -153310720\zeta(8) -3870720\zeta(2)\zeta(3)^2 -35078400\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+88429120\zeta(9) +28372800\zeta(3)\zeta(6) +45812160\zeta(4)\zeta(5) +18933120\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+1290240\zeta(3)^3 +149534919\zeta(10) -92839680\zeta(3)\zeta(7) +52912440\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.-6345360\zeta(2)\zeta(3)\zeta(5) -69396840\zeta(5)^2 -18196920 M(2,8) -9072000\zeta(2) M(2,6)\right) \tag{984} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k^{2}(k+1)^{2}} &= \frac{-1}{72}\left( 958192\zeta(8) +24192\zeta(2)\zeta(3)^2 +219240\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-545743\zeta(9) -177330\zeta(3)\zeta(6) -284436\zeta(4)\zeta(5) -118332\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-8064\zeta(3)^3\right) \tag{985} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k(k+1)^{3}} &= \frac{1}{23040}\left( 153310720\zeta(8) +3870720\zeta(2)\zeta(3)^2 +35078400\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-86208640\zeta(9) -28372800\zeta(3)\zeta(6) -45207360\zeta(4)\zeta(5) -18933120\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-1290240\zeta(3)^3 -149375151\zeta(10) +89769600\zeta(3)\zeta(7) -52206840\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+7514640\zeta(2)\zeta(3)\zeta(5) +67262760\zeta(5)^2 +17612280 M(2,8) +9072000\zeta(2) M(2,6)\right) \tag{986} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{(k+1)^{4}} &= \frac{1}{1152}\left( -16196565\zeta(11) -1630384\zeta(2)\zeta(9) -5721072\zeta(3)\zeta(8) \right. \nonumber \\ &\left.+10468728\zeta(4)\zeta(7) +10317144\zeta(5)\zeta(6) -837312\zeta(2)\zeta(3)^3 \right. \nonumber \\ &\left.+2330496\zeta(3)^2\zeta(5) +1411200\zeta(3) M(2,6) +616896 M(3,8)\right) \tag{987} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k^{3}(k+2)} &= \frac{-1}{46080}\left( -5760 -34560\zeta(2) -288000\zeta(3) -1546560\zeta(4) -4340160\zeta(5) \right. \nonumber \\ &\left.-927360\zeta(2)\zeta(3) -14115480\zeta(6) -1834560\zeta(3)^2 -12782160\zeta(7) -2923200\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-5957280\zeta(3)\zeta(4) -38327680\zeta(8) -967680\zeta(2)\zeta(3)^2 -8769600\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+44214560\zeta(9) +14186400\zeta(3)\zeta(6) +22906080\zeta(4)\zeta(5) +9466560\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+645120\zeta(3)^3 +149534919\zeta(10) -92839680\zeta(3)\zeta(7) +52912440\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.-6345360\zeta(2)\zeta(3)\zeta(5) -69396840\zeta(5)^2 -18196920 M(2,8) -9072000\zeta(2) M(2,6)\right) \tag{988} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{7}}{k^{2}(k+1)(k+2)} &= \frac{-1}{576}\left( -144 -864\zeta(2) -7200\zeta(3) -38664\zeta(4) -108504\zeta(5) \right. \nonumber \\ &\left.-23184\zeta(2)\zeta(3) -352887\zeta(6) -45864\zeta(3)^2 -319554\zeta(7) -73080\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-148932\zeta(3)\zeta(4) +2874576\zeta(8) +72576\zeta(2)\zeta(3)^2 +657720\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-1105364\zeta(9) -354660\zeta(3)\zeta(6) -572652\zeta(4)\zeta(5) -236664\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-16128\zeta(3)^3\right) \tag{989} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k(k+1)^{2}(k+2)} &= \frac{1}{288}\left( 144 +864\zeta(2) +7200\zeta(3) +38664\zeta(4) +108504\zeta(5) \right. \nonumber \\ &\left.+23184\zeta(2)\zeta(3) +352887\zeta(6) +45864\zeta(3)^2 +319554\zeta(7) +73080\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+148932\zeta(3)\zeta(4) +958192\zeta(8) +24192\zeta(2)\zeta(3)^2 +219240\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-1077608\zeta(9) -354660\zeta(3)\zeta(6) -565092\zeta(4)\zeta(5) -236664\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-16128\zeta(3)^3\right) \tag{990} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{(k+1)^{3}(k+2)} &= \frac{-1}{23040}\left( -23040 -138240\zeta(2) -1152000\zeta(3) -6186240\zeta(4) \right. \nonumber \\ &\left.-17360640\zeta(5) -3709440\zeta(2)\zeta(3) -56461920\zeta(6) -7338240\zeta(3)^2 -51128640\zeta(7) \right. \nonumber \\ &\left.-11692800\zeta(2)\zeta(5) -23829120\zeta(3)\zeta(4) +86208640\zeta(9) +28372800\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+45207360\zeta(4)\zeta(5) +18933120\zeta(2)\zeta(7) +1290240\zeta(3)^3 -149375151\zeta(10) \right. \nonumber \\ &\left.+89769600\zeta(3)\zeta(7) -52206840\zeta(3)^2\zeta(4) +7514640\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+67262760\zeta(5)^2 +17612280 M(2,8) +9072000\zeta(2) M(2,6)\right) \tag{991} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k^{2}(k+2)^{2}} &= \frac{1}{576}\left( -1296 -5904\zeta(2) -42192\zeta(3) -185004\zeta(4) -396216\zeta(5) \right. \nonumber \\ &\left.-87696\zeta(2)\zeta(3) -878271\zeta(6) -122472\zeta(3)^2 -288513\zeta(7) -59976\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-198072\zeta(3)\zeta(4) -243058\zeta(8) -63000\zeta(2)\zeta(3)^2 +256536\zeta(3)\zeta(5) +54936 M(2,6) \right. \nonumber \\ &\left.+1091486\zeta(9) +354660\zeta(3)\zeta(6) +568872\zeta(4)\zeta(5) +236664\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+16128\zeta(3)^3\right) \tag{992} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{k(k+1)(k+2)^{2}} &= \frac{1}{576}\left( -2736 -12672\zeta(2) -91584\zeta(3) -408672\zeta(4) -900936\zeta(5) \right. \nonumber \\ &\left.-198576\zeta(2)\zeta(3) -2109429\zeta(6) -290808\zeta(3)^2 -896580\zeta(7) -193032\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-545076\zeta(3)\zeta(4) +2388460\zeta(8) -53424\zeta(2)\zeta(3)^2 +1170792\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+109872 M(2,6) +1077608\zeta(9) +354660\zeta(3)\zeta(6) +565092\zeta(4)\zeta(5) +236664\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+16128\zeta(3)^3\right) \tag{993} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{(k+1)^{2}(k+2)^{2}} &= \frac{1}{144}\left( -1440 -6768\zeta(2) -49392\zeta(3) -223668\zeta(4) -504720\zeta(5) \right. \nonumber \\ &\left.-110880\zeta(2)\zeta(3) -1231158\zeta(6) -168336\zeta(3)^2 -608067\zeta(7) -133056\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-347004\zeta(3)\zeta(4) +715134\zeta(8) -38808\zeta(2)\zeta(3)^2 +475776\zeta(3)\zeta(5) +54936 M(2,6) \right. \nonumber \\ &\left.+1077608\zeta(9) +354660\zeta(3)\zeta(6) +565092\zeta(4)\zeta(5) +236664\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+16128\zeta(3)^3\right) \tag{994} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{7}}{k(k+2)^{3}} &= \frac{1}{46080}\left( 927360 +3179520\zeta(2) +19376640\zeta(3) +68423040\zeta(4) \right. \nonumber \\ &\left.+108054720\zeta(5) +24635520\zeta(2)\zeta(3) +140607960\zeta(6) +21107520\zeta(3)^2 -39775680\zeta(7) \right. \nonumber \\ &\left.-9979200\zeta(2)\zeta(5) -13134240\zeta(3)\zeta(4) -243547760\zeta(8) -19353600\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+16269120\zeta(3)\zeta(5) +12821760 M(2,6) -1803200\zeta(9) -98737440\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+16587360\zeta(4)\zeta(5) +3657600\zeta(2)\zeta(7) -4247040\zeta(3)^3 -149375151\zeta(10) \right. \nonumber \\ &\left.+89769600\zeta(3)\zeta(7) -52206840\zeta(3)^2\zeta(4) +7514640\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+67262760\zeta(5)^2 +17612280 M(2,8) +9072000\zeta(2) M(2,6)\right) \tag{995} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{(k+1)(k+2)^{3}} &= \frac{1}{23040}\left( 1036800 +3686400\zeta(2) +23040000\zeta(3) +84769920\zeta(4) \right. \nonumber \\ &\left.+144092160\zeta(5) +32578560\zeta(2)\zeta(3) +224985120\zeta(6) +32739840\zeta(3)^2 -3912480\zeta(7) \right. \nonumber \\ &\left.-2257920\zeta(2)\zeta(5) +8668800\zeta(3)\zeta(4) -339086160\zeta(8) -17216640\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-30562560\zeta(3)\zeta(5) +8426880 M(2,6) -44907520\zeta(9) -112923840\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-6016320\zeta(4)\zeta(5) -5808960\zeta(2)\zeta(7) -4892160\zeta(3)^3 -149375151\zeta(10) \right. \nonumber \\ &\left.+89769600\zeta(3)\zeta(7) -52206840\zeta(3)^2\zeta(4) +7514640\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+67262760\zeta(5)^2 +17612280 M(2,8) +9072000\zeta(2) M(2,6)\right) \tag{996} \\ \sum_{k=1}^\infty \frac{H(k)^{7}}{(k+2)^{4}} &= \frac{1}{1152}\left( -138240 -354816\zeta(2) -1838592\zeta(3) -5175072\zeta(4) -5821632\zeta(5) \right. \nonumber \\ &\left.-1338624\zeta(2)\zeta(3) -2929008\zeta(6) -443520\zeta(3)^2 +4509648\zeta(7) +959616\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+3108672\zeta(3)\zeta(4) +13269200\zeta(8) +1725696\zeta(2)\zeta(3)^2 -4491648\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-1314432 M(2,6) +327264\zeta(9) -101808\zeta(3)\zeta(6) -362880\zeta(4)\zeta(5) +145152\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+59136\zeta(3)^3 -16376535\zeta(10) +10304784\zeta(3)\zeta(7) -5156424\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+327600\zeta(2)\zeta(3)\zeta(5) +7137144\zeta(5)^2 +1989288 M(2,8) +947520\zeta(2) M(2,6) \right. \nonumber \\ &\left.-16196565\zeta(11) -1630384\zeta(2)\zeta(9) -5721072\zeta(3)\zeta(8) +10468728\zeta(4)\zeta(7) \right. \nonumber \\ &\left.+10317144\zeta(5)\zeta(6) -837312\zeta(2)\zeta(3)^3 +2330496\zeta(3)^2\zeta(5) +1411200\zeta(3) M(2,6) \right. \nonumber \\ &\left.+616896 M(3,8)\right) \tag{997} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{k^{3}} &= \frac{1}{72}\left( -2824380\zeta(11) -277304\zeta(2)\zeta(9) -1926401\zeta(3)\zeta(8) \right. \nonumber \\ &\left.+1998972\zeta(4)\zeta(7) +2270310\zeta(5)\zeta(6) -243648\zeta(2)\zeta(3)^3 +803808\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.+341280\zeta(3) M(2,6) +113760 M(3,8)\right) \tag{998} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{k^{2}(k+1)} &= \frac{1}{480}\left( -13336000\zeta(9) -7093200\zeta(3)\zeta(6) -6432000\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-2807280\zeta(2)\zeta(7) -322560\zeta(3)^3 +18741581\zeta(10) +6689520\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-524640\zeta(3)^2\zeta(4) +1452480\zeta(2)\zeta(3)\zeta(5) +4247040\zeta(5)^2 +485280 M(2,8) \right. \nonumber \\ &\left.+299520\zeta(2) M(2,6)\right) \tag{999} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{8}}{k(k+1)^{2}} &= \frac{-1}{240}\left( -6668000\zeta(9) -3546600\zeta(3)\zeta(6) -3216000\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-1403640\zeta(2)\zeta(7) -161280\zeta(3)^3 +9295879\zeta(10) +3314520\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-258540\zeta(3)^2\zeta(4) +733800\zeta(2)\zeta(3)\zeta(5) +2098980\zeta(5)^2 +238860 M(2,8) \right. \nonumber \\ &\left.+149760\zeta(2) M(2,6)\right) \tag{1000} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{(k+1)^{3}} &= \frac{-1}{72}\left( -2839707\zeta(11) -274424\zeta(2)\zeta(9) -1906367\zeta(3)\zeta(8) \right. \nonumber \\ &\left.+1976076\zeta(4)\zeta(7) +2252940\zeta(5)\zeta(6) -242208\zeta(2)\zeta(3)^3 +798912\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.+339120\zeta(3) M(2,6) +113040 M(3,8)\right) \tag{1001} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{k^{2}(k+2)} &= \frac{1}{2880}\left( -720 -5040\zeta(2) -49680\zeta(3) -324000\zeta(4) -1162800\zeta(5) \right. \nonumber \\ &\left.-247680\zeta(2)\zeta(3) -5303460\zeta(6) -692640\zeta(3)^2 -8496540\zeta(7) -1931040\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-4042800\zeta(3)\zeta(4) -19063670\zeta(8) -483840\zeta(2)\zeta(3)^2 -4368960\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-20004000\zeta(9) -10639800\zeta(3)\zeta(6) -9648000\zeta(4)\zeta(5) -4210920\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-483840\zeta(3)^3 +56224743\zeta(10) +20068560\zeta(3)\zeta(7) -1573920\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+4357440\zeta(2)\zeta(3)\zeta(5) +12741120\zeta(5)^2 +1455840 M(2,8) +898560\zeta(2) M(2,6)\right) \tag{1002} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{k(k+1)(k+2)} &= \frac{-1}{144}\left( 72 +504\zeta(2) +4968\zeta(3) +32400\zeta(4) +116280\zeta(5) \right. \nonumber \\ &\left.+24768\zeta(2)\zeta(3) +530346\zeta(6) +69264\zeta(3)^2 +849654\zeta(7) +193104\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+404280\zeta(3)\zeta(4) +1906367\zeta(8) +48384\zeta(2)\zeta(3)^2 +436896\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-2000400\zeta(9) -1063980\zeta(3)\zeta(6) -964800\zeta(4)\zeta(5) -421092\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-48384\zeta(3)^3\right) \tag{1003} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{(k+1)^{2}(k+2)} &= \frac{1}{720}\left( -720 -5040\zeta(2) -49680\zeta(3) -324000\zeta(4) -1162800\zeta(5) \right. \nonumber \\ &\left.-247680\zeta(2)\zeta(3) -5303460\zeta(6) -692640\zeta(3)^2 -8496540\zeta(7) -1931040\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-4042800\zeta(3)\zeta(4) -19063670\zeta(8) -483840\zeta(2)\zeta(3)^2 -4368960\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+27887637\zeta(10) +9943560\zeta(3)\zeta(7) -775620\zeta(3)^2\zeta(4) +2201400\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+6296940\zeta(5)^2 +716580 M(2,8) +449280\zeta(2) M(2,6)\right) \tag{1004} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{k(k+2)^{2}} &= \frac{1}{1440}\left( 6840 +37080\zeta(2) +317160\zeta(3) +1726560\zeta(4) +4930200\zeta(5) \right. \nonumber \\ &\left.+1074240\zeta(2)\zeta(3) +16758210\zeta(6) +2273040\zeta(3)^2 +16566750\zeta(7) \right. \nonumber \\ &\left.+3678480\zeta(2)\zeta(5) +8776440\zeta(3)\zeta(4) +14292825\zeta(8) +1501920\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-2962080\zeta(3)\zeta(5) -1098720 M(2,6) -11550160\zeta(9) -1773300\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-6477840\zeta(4)\zeta(5) -2627820\zeta(2)\zeta(7) -80640\zeta(3)^3 -27887637\zeta(10) \right. \nonumber \\ &\left.-9943560\zeta(3)\zeta(7) +775620\zeta(3)^2\zeta(4) -2201400\zeta(2)\zeta(3)\zeta(5) -6296940\zeta(5)^2 \right. \nonumber \\ &\left.-716580 M(2,8) -449280\zeta(2) M(2,6)\right) \tag{1005} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{8}}{(k+1)(k+2)^{2}} &= \frac{-1}{720}\left( -7200 -39600\zeta(2) -342000\zeta(3) -1888560\zeta(4) -5511600\zeta(5) \right. \nonumber \\ &\left.-1198080\zeta(2)\zeta(3) -19409940\zeta(6) -2619360\zeta(3)^2 -20815020\zeta(7) \right. \nonumber \\ &\left.-4644000\zeta(2)\zeta(5) -10797840\zeta(3)\zeta(4) -23824660\zeta(8) -1743840\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+777600\zeta(3)\zeta(5) +1098720 M(2,6) +21552160\zeta(9) +7093200\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+11301840\zeta(4)\zeta(5) +4733280\zeta(2)\zeta(7) +322560\zeta(3)^3 +27887637\zeta(10) \right. \nonumber \\ &\left.+9943560\zeta(3)\zeta(7) -775620\zeta(3)^2\zeta(4) +2201400\zeta(2)\zeta(3)\zeta(5) +6296940\zeta(5)^2 \right. \nonumber \\ &\left.+716580 M(2,8) +449280\zeta(2) M(2,6)\right) \tag{1006} \\ \sum_{k=1}^\infty \frac{H(k)^{8}}{(k+2)^{3}} &= \frac{1}{2880}\left( -129600 -541440\zeta(2) -4008960\zeta(3) -18103680\zeta(4) -40584960\zeta(5) \right. \nonumber \\ &\left.-9020160\zeta(2)\zeta(3) -98753280\zeta(6) -13881600\zeta(3)^2 -48610080\zeta(7) \right. \nonumber \\ &\left.-10391040\zeta(2)\zeta(5) -28817280\zeta(3)\zeta(4) +74914520\zeta(8) +1733760\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+18086400\zeta(3)\zeta(5) +120960 M(2,6) +65558080\zeta(9) +70648320\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+25611840\zeta(4)\zeta(5) +12371040\zeta(2)\zeta(7) +3091200\zeta(3)^3 +149375151\zeta(10) \right. \nonumber \\ &\left.-89769600\zeta(3)\zeta(7) +52206840\zeta(3)^2\zeta(4) -7514640\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-67262760\zeta(5)^2 -17612280 M(2,8) -9072000\zeta(2) M(2,6) +113588280\zeta(11) \right. \nonumber \\ &\left.+10976960\zeta(2)\zeta(9) +76254680\zeta(3)\zeta(8) -79043040\zeta(4)\zeta(7) -90117600\zeta(5)\zeta(6) \right. \nonumber \\ &\left.+9688320\zeta(2)\zeta(3)^3 -31956480\zeta(3)^2\zeta(5) -13564800\zeta(3) M(2,6) -4521600 M(3,8)\right) \tag{1007} \\ \sum_{k=1}^\infty \frac{H(k)^{9}}{k^{2}} &= \frac{-1}{64}\left( -7739347\zeta(11) -2048432\zeta(2)\zeta(9) -5357920\zeta(3)\zeta(8) \right. \nonumber \\ &\left.-8811792\zeta(4)\zeta(7) -10526056\zeta(5)\zeta(6) +294208\zeta(2)\zeta(3)^3 -2064192\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.-540096\zeta(3) M(2,6) -199936 M(3,8)\right) \tag{1008} \\ \sum_{k=1}^\infty \frac{H(k)^{9}}{k(k+1)} &= \frac{1}{40}\left( 17039209\zeta(10) +3158190\zeta(3)\zeta(7) +704820\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+928080\zeta(2)\zeta(3)\zeta(5) +1767000\zeta(5)^2 +37320\zeta(2) M(2,6)\right) \tag{1009} \\ \sum_{k=1}^\infty \frac{H(k)^{9}}{(k+1)^{2}} &= \frac{1}{64}\left( 7676163\zeta(11) +2050992\zeta(2)\zeta(9) +5357920\zeta(3)\zeta(8) \right. \nonumber \\ &\left.+8776416\zeta(4)\zeta(7) +10489496\zeta(5)\zeta(6) -292928\zeta(2)\zeta(3)^3 +2058432\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.+538176\zeta(3) M(2,6) +199296 M(3,8)\right) \tag{1010} \\ \sum_{k=1}^\infty \frac{H(k)^{9}}{k(k+2)} &= \frac{1}{160}\left( 80 +640\zeta(2) +7280\zeta(3) +55780\zeta(4) +243200\zeta(5) \right. \nonumber \\ &\left.+51600\zeta(2)\zeta(3) +1416700\zeta(6) +185160\zeta(3)^2 +3186110\zeta(7) +721920\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+1525680\zeta(3)\zeta(4) +12880535\zeta(8) +365400\zeta(2)\zeta(3)^2 +2739360\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-37320 M(2,6) +9964440\zeta(9) +5312700\zeta(3)\zeta(6) +4812120\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+2105460\zeta(2)\zeta(7) +241760\zeta(3)^3 +34078418\zeta(10) +6316380\zeta(3)\zeta(7) \right. \nonumber \\ &\left.+1409640\zeta(3)^2\zeta(4) +1856160\zeta(2)\zeta(3)\zeta(5) +3534000\zeta(5)^2 \right. \nonumber \\ &\left.+74640\zeta(2) M(2,6)\right) \tag{1011} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{9}}{(k+1)(k+2)} &= \frac{-1}{16}\left( -16 -128\zeta(2) -1456\zeta(3) -11156\zeta(4) -48640\zeta(5) \right. \nonumber \\ &\left.-10320\zeta(2)\zeta(3) -283340\zeta(6) -37032\zeta(3)^2 -637222\zeta(7) -144384\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-305136\zeta(3)\zeta(4) -2576107\zeta(8) -73080\zeta(2)\zeta(3)^2 -547872\zeta(3)\zeta(5) +7464 M(2,6) \right. \nonumber \\ &\left.-1992888\zeta(9) -1062540\zeta(3)\zeta(6) -962424\zeta(4)\zeta(5) -421092\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-48352\zeta(3)^3\right) \tag{1012} \\ \sum_{k=1}^\infty \frac{H(k)^{9}}{(k+2)^{2}} &= \frac{-1}{320}\left( 3200 +20160\zeta(2) +200960\zeta(3) +1307040\zeta(4) +4662400\zeta(5) \right. \nonumber \\ &\left.+1005120\zeta(2)\zeta(3) +21224080\zeta(6) +2841600\zeta(3)^2 +33558720\zeta(7) \right. \nonumber \\ &\left.+7513920\zeta(2)\zeta(5) +16977600\zeta(3)\zeta(4) +83974040\zeta(8) +3958080\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+9227520\zeta(3)\zeta(5) -1763520 M(2,6) -3246560\zeta(9) +7064400\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-3355200\zeta(4)\zeta(5) -1044720\zeta(2)\zeta(7) +321920\zeta(3)^3 -111550548\zeta(10) \right. \nonumber \\ &\left.-39774240\zeta(3)\zeta(7) +3102480\zeta(3)^2\zeta(4) -8805600\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-25187760\zeta(5)^2 -2866320 M(2,8) -1797120\zeta(2) M(2,6) -38380815\zeta(11) \right. \nonumber \\ &\left.-10254960\zeta(2)\zeta(9) -26789600\zeta(3)\zeta(8) -43882080\zeta(4)\zeta(7) -52447480\zeta(5)\zeta(6) \right. \nonumber \\ &\left.+1464640\zeta(2)\zeta(3)^3 -10292160\zeta(3)^2\zeta(5) -2690880\zeta(3) M(2,6) -996480 M(3,8)\right) \tag{1013} \end{align}\]

Formulas for order \(r = m + n = 12\): \[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{11}} &= \frac{-1}{4}\left( -13\zeta(12) +4\zeta(3)\zeta(9) +4\zeta(5)\zeta(7)\right) \tag{1014} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{10}(k+1)} &= \frac{-1}{4}\left( 4\zeta(2) -8\zeta(3) +5\zeta(4) -12\zeta(5) +4\zeta(2)\zeta(3) +7\zeta(6) \right. \nonumber \\ &\left.-2\zeta(3)^2 -16\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4) +9\zeta(8) -4\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-20\zeta(9) +4\zeta(3)\zeta(6) +4\zeta(4)\zeta(5) +4\zeta(2)\zeta(7) +11\zeta(10) -4\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-2\zeta(5)^2 -24\zeta(11) +4\zeta(2)\zeta(9) +4\zeta(3)\zeta(8) +4\zeta(4)\zeta(7) \right. \nonumber \\ &\left.+4\zeta(5)\zeta(6)\right) \tag{1015} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{9}(k+1)^{2}} &= \frac{-1}{4}\left( -36\zeta(2) +68\zeta(3) -35\zeta(4) +72\zeta(5) -24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-35\zeta(6) +10\zeta(3)^2 +64\zeta(7) -16\zeta(2)\zeta(5) -16\zeta(3)\zeta(4) -27\zeta(8) \right. \nonumber \\ &\left.+12\zeta(3)\zeta(5) +40\zeta(9) -8\zeta(3)\zeta(6) -8\zeta(4)\zeta(5) -8\zeta(2)\zeta(7) -11\zeta(10) \right. \nonumber \\ &\left.+4\zeta(3)\zeta(7) +2\zeta(5)^2\right) \tag{1016} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{8}(k+1)^{3}} &= \frac{1}{4}\left( -144\zeta(2) +256\zeta(3) -104\zeta(4) +180\zeta(5) -60\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-70\zeta(6) +20\zeta(3)^2 +96\zeta(7) -24\zeta(2)\zeta(5) -24\zeta(3)\zeta(4) -27\zeta(8) \right. \nonumber \\ &\left.+12\zeta(3)\zeta(5) +20\zeta(9) -4\zeta(3)\zeta(6) -4\zeta(4)\zeta(5) -4\zeta(2)\zeta(7)\right) \tag{1017} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{7}(k+1)^{4}} &= \frac{-1}{4}\left( -336\zeta(2) +560\zeta(3) -168\zeta(4) +248\zeta(5) -84\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-70\zeta(6) +20\zeta(3)^2 +64\zeta(7) -16\zeta(2)\zeta(5) -16\zeta(3)\zeta(4) -9\zeta(8) \right. \nonumber \\ &\left.+4\zeta(3)\zeta(5)\right) \tag{1018} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{6}(k+1)^{5}} &= \frac{1}{2}\left( -252\zeta(2) +392\zeta(3) -77\zeta(4) +114\zeta(5) -42\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-16\zeta(6) +4\zeta(3)^2 +8\zeta(7) -2\zeta(2)\zeta(5) -2\zeta(3)\zeta(4)\right) \tag{1019} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{5}(k+1)^{6}} &= \frac{-1}{2}\left( -252\zeta(2) +364\zeta(3) -35\zeta(4) +96\zeta(5) -42\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+4\zeta(6) -4\zeta(3)^2 +6\zeta(7) -2\zeta(2)\zeta(5) -2\zeta(3)\zeta(4)\right) \tag{1020} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{4}(k+1)^{7}} &= \frac{-1}{4}\left( 336\zeta(2) -448\zeta(3) -172\zeta(5) +84\zeta(2)\zeta(3) -30\zeta(6) \right. \nonumber \\ &\left.+20\zeta(3)^2 -48\zeta(7) +16\zeta(2)\zeta(5) +16\zeta(3)\zeta(4) -5\zeta(8) +4\zeta(3)\zeta(5)\right) \tag{1021} \\ \sum_{k=1}^\infty \frac{H(k)}{k^{3}(k+1)^{8}} &= \frac{-1}{4}\left( -144\zeta(2) +176\zeta(3) +16\zeta(4) +120\zeta(5) -60\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+30\zeta(6) -20\zeta(3)^2 +72\zeta(7) -24\zeta(2)\zeta(5) -24\zeta(3)\zeta(4) +15\zeta(8) \right. \nonumber \\ &\left.-12\zeta(3)\zeta(5) +16\zeta(9) -4\zeta(3)\zeta(6) -4\zeta(4)\zeta(5) -4\zeta(2)\zeta(7)\right) \tag{1022} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)}{k^{2}(k+1)^{9}} &= \frac{-1}{4}\left( 36\zeta(2) -40\zeta(3) -7\zeta(4) -48\zeta(5) +24\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-15\zeta(6) +10\zeta(3)^2 -48\zeta(7) +16\zeta(2)\zeta(5) +16\zeta(3)\zeta(4) -15\zeta(8) \right. \nonumber \\ &\left.+12\zeta(3)\zeta(5) -32\zeta(9) +8\zeta(3)\zeta(6) +8\zeta(4)\zeta(5) +8\zeta(2)\zeta(7) -7\zeta(10) \right. \nonumber \\ &\left.+4\zeta(3)\zeta(7) +2\zeta(5)^2\right) \tag{1023} \\ \sum_{k=1}^\infty \frac{H(k)}{k(k+1)^{10}} &= \frac{1}{4}\left( 4\zeta(2) -4\zeta(3) -\zeta(4) -8\zeta(5) +4\zeta(2)\zeta(3) -3\zeta(6) \right. \nonumber \\ &\left.+2\zeta(3)^2 -12\zeta(7) +4\zeta(2)\zeta(5) +4\zeta(3)\zeta(4) -5\zeta(8) +4\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-16\zeta(9) +4\zeta(3)\zeta(6) +4\zeta(4)\zeta(5) +4\zeta(2)\zeta(7) -7\zeta(10) +4\zeta(3)\zeta(7) \right. \nonumber \\ &\left.+2\zeta(5)^2 -20\zeta(11) +4\zeta(2)\zeta(9) +4\zeta(3)\zeta(8) +4\zeta(4)\zeta(7) \right. \nonumber \\ &\left.+4\zeta(5)\zeta(6)\right) \tag{1024} \\ \sum_{k=1}^\infty \frac{H(k)}{(k+1)^{11}} &= \frac{-1}{4}\left( -9\zeta(12) +4\zeta(3)\zeta(9) +4\zeta(5)\zeta(7)\right) \tag{1025} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{10}} &= -\left( - M(2,10)\right) \tag{1026} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{9}(k+1)} &= \frac{1}{24}\left( 72\zeta(3) -102\zeta(4) +84\zeta(5) -24\zeta(2)\zeta(3) -97\zeta(6) \right. \nonumber \\ &\left.+48\zeta(3)^2 +144\zeta(7) -24\zeta(2)\zeta(5) -60\zeta(3)\zeta(4) -24 M(2,6) +220\zeta(9) \right. \nonumber \\ &\left.-84\zeta(3)\zeta(6) -60\zeta(4)\zeta(5) -24\zeta(2)\zeta(7) +8\zeta(3)^3 -24 M(2,8) +312\zeta(11) \right. \nonumber \\ &\left.-24\zeta(2)\zeta(9) -108\zeta(3)\zeta(8) -60\zeta(4)\zeta(7) -84\zeta(5)\zeta(6) \right. \nonumber \\ &\left.+24\zeta(3)^2\zeta(5)\right) \tag{1027} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{8}(k+1)^{2}} &= \frac{1}{24}\left( -576\zeta(3) +780\zeta(4) -504\zeta(5) +144\zeta(2)\zeta(3) +485\zeta(6) \right. \nonumber \\ &\left.-240\zeta(3)^2 -576\zeta(7) +96\zeta(2)\zeta(5) +240\zeta(3)\zeta(4) +72 M(2,6) -440\zeta(9) \right. \nonumber \\ &\left.+168\zeta(3)\zeta(6) +120\zeta(4)\zeta(5) +48\zeta(2)\zeta(7) -16\zeta(3)^3 +24 M(2,8)\right) \tag{1028} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{7}(k+1)^{3}} &= \frac{-1}{12}\left( -1008\zeta(3) +1302\zeta(4) -648\zeta(5) +192\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+485\zeta(6) -240\zeta(3)^2 -432\zeta(7) +72\zeta(2)\zeta(5) +180\zeta(3)\zeta(4) +36 M(2,6) \right. \nonumber \\ &\left.-110\zeta(9) +42\zeta(3)\zeta(6) +30\zeta(4)\zeta(5) +12\zeta(2)\zeta(7) -4\zeta(3)^3\right) \tag{1029} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{6}(k+1)^{4}} &= \frac{1}{24}\left( -4032\zeta(3) +4956\zeta(4) -1896\zeta(5) +624\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+1007\zeta(6) -504\zeta(3)^2 -576\zeta(7) +96\zeta(2)\zeta(5) +240\zeta(3)\zeta(4) +24 M(2,6)\right) \tag{1030} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{5}(k+1)^{5}} &= \frac{1}{12}\left( 2520\zeta(3) -2940\zeta(4) +900\zeta(5) -360\zeta(2)\zeta(3) \right. \nonumber \\ &\left.-335\zeta(6) +180\zeta(3)^2 +84\zeta(7) -24\zeta(2)\zeta(5) -24\zeta(3)\zeta(4)\right) \tag{1031} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{4}(k+1)^{6}} &= \frac{1}{24}\left( -4032\zeta(3) +4452\zeta(4) -1224\zeta(5) +624\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+467\zeta(6) -288\zeta(3)^2 -96\zeta(7) +96\zeta(2)\zeta(5) -48\zeta(3)\zeta(4) -84\zeta(8) \right. \nonumber \\ &\left.+48\zeta(3)\zeta(5) +24 M(2,6)\right) \tag{1032} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{3}(k+1)^{7}} &= \frac{-1}{12}\left( -1008\zeta(3) +1050\zeta(4) -312\zeta(5) +192\zeta(2)\zeta(3) \right. \nonumber \\ &\left.+185\zeta(6) -120\zeta(3)^2 -72\zeta(7) +72\zeta(2)\zeta(5) -36\zeta(3)\zeta(4) -126\zeta(8) \right. \nonumber \\ &\left.+72\zeta(3)\zeta(5) +36 M(2,6) +2\zeta(9) -18\zeta(3)\zeta(6) -6\zeta(4)\zeta(5) +12\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+4\zeta(3)^3\right) \tag{1033} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k^{2}(k+1)^{8}} &= \frac{1}{24}\left( -576\zeta(3) +564\zeta(4) -216\zeta(5) +144\zeta(2)\zeta(3) +185\zeta(6) \right. \nonumber \\ &\left.-120\zeta(3)^2 -96\zeta(7) +96\zeta(2)\zeta(5) -48\zeta(3)\zeta(4) -252\zeta(8) +144\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+72 M(2,6) +8\zeta(9) -72\zeta(3)\zeta(6) -24\zeta(4)\zeta(5) +48\zeta(2)\zeta(7) +16\zeta(3)^3 \right. \nonumber \\ &\left.-108\zeta(10) +48\zeta(3)\zeta(7) +24\zeta(5)^2 +24 M(2,8)\right) \tag{1034} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{k(k+1)^{9}} &= \frac{1}{24}\left( 72\zeta(3) -66\zeta(4) +36\zeta(5) -24\zeta(2)\zeta(3) -37\zeta(6) \right. \nonumber \\ &\left.+24\zeta(3)^2 +24\zeta(7) -24\zeta(2)\zeta(5) +12\zeta(3)\zeta(4) +84\zeta(8) -48\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-24 M(2,6) -4\zeta(9) +36\zeta(3)\zeta(6) +12\zeta(4)\zeta(5) -24\zeta(2)\zeta(7) -8\zeta(3)^3 \right. \nonumber \\ &\left.+108\zeta(10) -48\zeta(3)\zeta(7) -24\zeta(5)^2 -24 M(2,8) -48\zeta(11) -24\zeta(2)\zeta(9) \right. \nonumber \\ &\left.+60\zeta(3)\zeta(8) +12\zeta(4)\zeta(7) +36\zeta(5)\zeta(6) -24\zeta(3)^2\zeta(5)\right) \tag{1035} \\ \sum_{k=1}^\infty \frac{H(k)^{2}}{(k+1)^{10}} &= \frac{-1}{2}\left( 11\zeta(12) -4\zeta(3)\zeta(9) -4\zeta(5)\zeta(7) -2 M(2,10)\right) \tag{1036} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{9}} &= \frac{-1}{22112}\left( 355355\zeta(12) -221120\zeta(3)\zeta(9) -265344\zeta(5)\zeta(7) \right. \nonumber \\ &\left.-33168\zeta(3)^2\zeta(6) +5528\zeta(3)^4 +49752\zeta(2)\zeta(5)^2 +99504\zeta(2)\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-82920 M(2,10)\right) \tag{1037} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{8}(k+1)} &= \frac{-1}{480}\left( 4800\zeta(4) -4800\zeta(5) -480\zeta(2)\zeta(3) +2790\zeta(6) \right. \nonumber \\ &\left.-1200\zeta(3)^2 -6930\zeta(7) -960\zeta(2)\zeta(5) +6120\zeta(3)\zeta(4) -2975\zeta(8) \right. \nonumber \\ &\left.-600\zeta(2)\zeta(3)^2 +2880\zeta(3)\zeta(5) +1320 M(2,6) -10420\zeta(9) +5820\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+6120\zeta(4)\zeta(5) -1440\zeta(2)\zeta(7) -960\zeta(3)^3 -4983\zeta(10) +3840\zeta(3)\zeta(7) \right. \nonumber \\ &\left.+240\zeta(3)^2\zeta(4) -1680\zeta(2)\zeta(3)\zeta(5) +2160\zeta(5)^2 +1560 M(2,8) -480 M(3,8)\right) \tag{1038} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{7}(k+1)^{2}} &= \frac{1}{480}\left( 33600\zeta(4) -32400\zeta(5) -3360\zeta(2)\zeta(3) +13950\zeta(6) \right. \nonumber \\ &\left.-6000\zeta(3)^2 -27720\zeta(7) -3840\zeta(2)\zeta(5) +24480\zeta(3)\zeta(4) -8925\zeta(8) \right. \nonumber \\ &\left.-1800\zeta(2)\zeta(3)^2 +8640\zeta(3)\zeta(5) +3960 M(2,6) -20840\zeta(9) +11640\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+12240\zeta(4)\zeta(5) -2880\zeta(2)\zeta(7) -1920\zeta(3)^3 -4983\zeta(10) +3840\zeta(3)\zeta(7) \right. \nonumber \\ &\left.+240\zeta(3)^2\zeta(4) -1680\zeta(2)\zeta(3)\zeta(5) +2160\zeta(5)^2 +1560 M(2,8)\right) \tag{1039} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{6}(k+1)^{3}} &= \frac{1}{96}\left( -20160\zeta(4) +18720\zeta(5) +2016\zeta(2)\zeta(3) -5778\zeta(6) \right. \nonumber \\ &\left.+2592\zeta(3)^2 +8316\zeta(7) +1152\zeta(2)\zeta(5) -7344\zeta(3)\zeta(4) +1785\zeta(8) \right. \nonumber \\ &\left.+360\zeta(2)\zeta(3)^2 -1728\zeta(3)\zeta(5) -792 M(2,6) +2084\zeta(9) -1164\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-1224\zeta(4)\zeta(5) +288\zeta(2)\zeta(7) +192\zeta(3)^3\right) \tag{1040} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{5}(k+1)^{4}} &= \frac{-1}{96}\left( -33600\zeta(4) +30000\zeta(5) +3360\zeta(2)\zeta(3) -6570\zeta(6) \right. \nonumber \\ &\left.+3360\zeta(3)^2 +6258\zeta(7) +960\zeta(2)\zeta(5) -5688\zeta(3)\zeta(4) +595\zeta(8) \right. \nonumber \\ &\left.+120\zeta(2)\zeta(3)^2 -576\zeta(3)\zeta(5) -264 M(2,6)\right) \tag{1041} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{4}(k+1)^{5}} &= \frac{1}{96}\left( -33600\zeta(4) +28800\zeta(5) +3360\zeta(2)\zeta(3) -4770\zeta(6) \right. \nonumber \\ &\left.+3120\zeta(3)^2 +4242\zeta(7) +960\zeta(2)\zeta(5) -4392\zeta(3)\zeta(4) -43\zeta(8) \right. \nonumber \\ &\left.-120\zeta(2)\zeta(3)^2 +288\zeta(3)\zeta(5) -24 M(2,6)\right) \tag{1042} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{3}(k+1)^{6}} &= \frac{1}{96}\left( 20160\zeta(4) -16560\zeta(5) -2016\zeta(2)\zeta(3) +2538\zeta(6) \right. \nonumber \\ &\left.-2160\zeta(3)^2 -4284\zeta(7) -1152\zeta(2)\zeta(5) +4752\zeta(3)\zeta(4) +129\zeta(8) \right. \nonumber \\ &\left.+360\zeta(2)\zeta(3)^2 -864\zeta(3)\zeta(5) +72 M(2,6) -788\zeta(9) +444\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+792\zeta(4)\zeta(5) -288\zeta(2)\zeta(7) -96\zeta(3)^3\right) \tag{1043} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k^{2}(k+1)^{7}} &= \frac{1}{480}\left( -33600\zeta(4) +26400\zeta(5) +3360\zeta(2)\zeta(3) -4950\zeta(6) \right. \nonumber \\ &\left.+4800\zeta(3)^2 +14280\zeta(7) +3840\zeta(2)\zeta(5) -15840\zeta(3)\zeta(4) -645\zeta(8) \right. \nonumber \\ &\left.-1800\zeta(2)\zeta(3)^2 +4320\zeta(3)\zeta(5) -360 M(2,6) +7880\zeta(9) -4440\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-7920\zeta(4)\zeta(5) +2880\zeta(2)\zeta(7) +960\zeta(3)^3 -1503\zeta(10) +2400\zeta(3)\zeta(7) \right. \nonumber \\ &\left.+240\zeta(3)^2\zeta(4) -1680\zeta(2)\zeta(3)\zeta(5) +1440\zeta(5)^2 +120 M(2,8)\right) \tag{1044} \\ \sum_{k=1}^\infty \frac{H(k)^{3}}{k(k+1)^{8}} &= \frac{1}{480}\left( 4800\zeta(4) -3600\zeta(5) -480\zeta(2)\zeta(3) +990\zeta(6) -960\zeta(3)^2 \right. \nonumber \\ &\left.-3570\zeta(7) -960\zeta(2)\zeta(5) +3960\zeta(3)\zeta(4) +215\zeta(8) +600\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-1440\zeta(3)\zeta(5) +120 M(2,6) -3940\zeta(9) +2220\zeta(3)\zeta(6) +3960\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-1440\zeta(2)\zeta(7) -480\zeta(3)^3 +1503\zeta(10) -2400\zeta(3)\zeta(7) -240\zeta(3)^2\zeta(4) \right. \nonumber \\ &\left.+1680\zeta(2)\zeta(3)\zeta(5) -1440\zeta(5)^2 -120 M(2,8) +10560\zeta(11) -5040\zeta(3)\zeta(8) \right. \nonumber \\ &\left.-2160\zeta(4)\zeta(7) -3600\zeta(5)\zeta(6) +1440\zeta(3)^2\zeta(5) -480 M(3,8)\right) \tag{1045} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{3}}{(k+1)^{9}} &= \frac{1}{22112}\left( -161875\zeta(12) +154784\zeta(3)\zeta(9) +199008\zeta(5)\zeta(7) \right. \nonumber \\ &\left.+33168\zeta(3)^2\zeta(6) -5528\zeta(3)^4 -49752\zeta(2)\zeta(5)^2 -99504\zeta(2)\zeta(3)\zeta(7) \right. \nonumber \\ &\left.+16584 M(2,10)\right) \tag{1046} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{8}} &= -\left( - M(4,8)\right) \tag{1047} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{7}(k+1)} &= \frac{1}{5760}\left( 172800\zeta(5) +34560\zeta(2)\zeta(3) -234960\zeta(6) -17280\zeta(3)^2 \right. \nonumber \\ &\left.+133200\zeta(7) +28800\zeta(2)\zeta(5) -123840\zeta(3)\zeta(4) +593320\zeta(8) \right. \nonumber \\ &\left.+161280\zeta(2)\zeta(3)^2 -668160\zeta(3)\zeta(5) -149760 M(2,6) +209280\zeta(9) \right. \nonumber \\ &\left.-133920\zeta(3)\zeta(6) -123840\zeta(4)\zeta(5) +40320\zeta(2)\zeta(7) +19200\zeta(3)^3 \right. \nonumber \\ &\left.+619407\zeta(10) -540000\zeta(3)\zeta(7) -9000\zeta(3)^2\zeta(4) +195120\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-212040\zeta(5)^2 -109080 M(2,8) -11520\zeta(2) M(2,6) -345240\zeta(11) -32640\zeta(2)\zeta(9) \right. \nonumber \\ &\left.+142800\zeta(3)\zeta(8) +145440\zeta(4)\zeta(7) +122160\zeta(5)\zeta(6) +9600\zeta(2)\zeta(3)^3 \right. \nonumber \\ &\left.-69120\zeta(3)^2\zeta(5) +21120 M(3,8)\right) \tag{1048} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{6}(k+1)^{2}} &= \frac{-1}{1920}\left( 345600\zeta(5) +69120\zeta(2)\zeta(3) -460320\zeta(6) \right. \nonumber \\ &\left.-34560\zeta(3)^2 +177600\zeta(7) +38400\zeta(2)\zeta(5) -165120\zeta(3)\zeta(4) +593320\zeta(8) \right. \nonumber \\ &\left.+161280\zeta(2)\zeta(3)^2 -668160\zeta(3)\zeta(5) -149760 M(2,6) +139520\zeta(9) -89280\zeta(3)\zeta(6) \right. \nonumber \\ &\left.-82560\zeta(4)\zeta(5) +26880\zeta(2)\zeta(7) +12800\zeta(3)^3 +206469\zeta(10) -180000\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-3000\zeta(3)^2\zeta(4) +65040\zeta(2)\zeta(3)\zeta(5) -70680\zeta(5)^2 -36360 M(2,8) \right. \nonumber \\ &\left.-3840\zeta(2) M(2,6)\right) \tag{1049} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{5}(k+1)^{3}} &= \frac{-1}{48}\left( -21600\zeta(5) -4320\zeta(2)\zeta(3) +28170\zeta(6) +2160\zeta(3)^2 \right. \nonumber \\ &\left.-7314\zeta(7) -1680\zeta(2)\zeta(5) +7080\zeta(3)\zeta(4) -14833\zeta(8) -4032\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+16704\zeta(3)\zeta(5) +3744 M(2,6) -1744\zeta(9) +1116\zeta(3)\zeta(6) +1032\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-336\zeta(2)\zeta(7) -160\zeta(3)^3\right) \tag{1050} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{4}(k+1)^{4}} &= \frac{1}{18}\left( -10800\zeta(5) -2160\zeta(2)\zeta(3) +13785\zeta(6) +1080\zeta(3)^2 \right. \nonumber \\ &\left.-2646\zeta(7) -720\zeta(2)\zeta(5) +2880\zeta(3)\zeta(4) -3406\zeta(8) -918\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+3816\zeta(3)\zeta(5) +846 M(2,6)\right) \tag{1051} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{3}(k+1)^{5}} &= \frac{1}{48}\left( 21600\zeta(5) +4320\zeta(2)\zeta(3) -26970\zeta(6) -2160\zeta(3)^2 \right. \nonumber \\ &\left.+5034\zeta(7) +1680\zeta(2)\zeta(5) -6360\zeta(3)\zeta(4) +12415\zeta(8) +3312\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-13824\zeta(3)\zeta(5) -3024 M(2,6) +696\zeta(9) -396\zeta(3)\zeta(6) -888\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+336\zeta(2)\zeta(7) +128\zeta(3)^3\right) \tag{1052} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{4}}{k^{2}(k+1)^{6}} &= \frac{1}{1920}\left( -345600\zeta(5) -69120\zeta(2)\zeta(3) +421920\zeta(6) +34560\zeta(3)^2 \right. \nonumber \\ &\left.-104640\zeta(7) -38400\zeta(2)\zeta(5) +142080\zeta(3)\zeta(4) -496600\zeta(8) \right. \nonumber \\ &\left.-132480\zeta(2)\zeta(3)^2 +552960\zeta(3)\zeta(5) +120960 M(2,6) -55680\zeta(9) +31680\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+71040\zeta(4)\zeta(5) -26880\zeta(2)\zeta(7) -10240\zeta(3)^3 -145941\zeta(10) +126240\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-840\zeta(3)^2\zeta(4) -38160\zeta(2)\zeta(3)\zeta(5) +39960\zeta(5)^2 +22920 M(2,8) \right. \nonumber \\ &\left.+3840\zeta(2) M(2,6)\right) \tag{1053} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{k(k+1)^{7}} &= \frac{-1}{5760}\left( -172800\zeta(5) -34560\zeta(2)\zeta(3) +206160\zeta(6) +17280\zeta(3)^2 \right. \nonumber \\ &\left.-78480\zeta(7) -28800\zeta(2)\zeta(5) +106560\zeta(3)\zeta(4) -496600\zeta(8) -132480\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+552960\zeta(3)\zeta(5) +120960 M(2,6) -83520\zeta(9) +47520\zeta(3)\zeta(6) +106560\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-40320\zeta(2)\zeta(7) -15360\zeta(3)^3 -437823\zeta(10) +378720\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-2520\zeta(3)^2\zeta(4) -114480\zeta(2)\zeta(3)\zeta(5) +119880\zeta(5)^2 +68760 M(2,8) \right. \nonumber \\ &\left.+11520\zeta(2) M(2,6) -28440\zeta(11) -44160\zeta(2)\zeta(9) +10320\zeta(3)\zeta(8) \right. \nonumber \\ &\left.+82080\zeta(4)\zeta(7) +24240\zeta(5)\zeta(6) +9600\zeta(2)\zeta(3)^3 -34560\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.-1920 M(3,8)\right) \tag{1054} \\ \sum_{k=1}^\infty \frac{H(k)^{4}}{(k+1)^{8}} &= \frac{-1}{5528}\left( -289019\zeta(12) +199008\zeta(3)\zeta(9) +243232\zeta(5)\zeta(7) \right. \nonumber \\ &\left.+33168\zeta(3)^2\zeta(6) -5528\zeta(3)^4 -49752\zeta(2)\zeta(5)^2 -99504\zeta(2)\zeta(3)\zeta(7) \right. \nonumber \\ &\left.+49752 M(2,10) -5528 M(4,8)\right) \tag{1055} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{7}} &= \frac{1}{265344}\left( 3612841\zeta(12) -884480\zeta(3)\zeta(9) -597024\zeta(5)\zeta(7) \right. \nonumber \\ &\left.+364848\zeta(3)^2\zeta(6) +221120\zeta(3)^4 +364848\zeta(2)\zeta(5)^2 +729696\zeta(2)\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-3250464\zeta(4)\zeta(3)\zeta(5) -1028208 M(2,10) +663360 M(4,8)\right) \tag{1056} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{6}(k+1)} &= \frac{-1}{2304}\left( 411264\zeta(6) +51840\zeta(3)^2 -295344\zeta(7) -65664\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-76032\zeta(3)\zeta(4) -542488\zeta(8) -152640\zeta(2)\zeta(3)^2 +630144\zeta(3)\zeta(5) +135360 M(2,6) \right. \nonumber \\ &\left.-302144\zeta(9) +469920\zeta(3)\zeta(6) -152064\zeta(4)\zeta(5) -76320\zeta(2)\zeta(7) +11520\zeta(3)^3 \right. \nonumber \\ &\left.-579897\zeta(10) +519840\zeta(3)\zeta(7) -3240\zeta(3)^2\zeta(4) -185040\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+203832\zeta(5)^2 +98280 M(2,8) +11520\zeta(2) M(2,6) +3126684\zeta(11) +352064\zeta(2)\zeta(9) \right. \nonumber \\ &\left.-1186640\zeta(3)\zeta(8) -1647936\zeta(4)\zeta(7) -880320\zeta(5)\zeta(6) -84480\zeta(2)\zeta(3)^3 \right. \nonumber \\ &\left.+564480\zeta(3)^2\zeta(5) -34560\zeta(3) M(2,6) -111360 M(3,8)\right) \tag{1057} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{5}(k+1)^{2}} &= \frac{1}{2304}\left( 2056320\zeta(6) +259200\zeta(3)^2 -1448496\zeta(7) \right. \nonumber \\ &\left.-328320\zeta(2)\zeta(5) -380160\zeta(3)\zeta(4) -1627464\zeta(8) -457920\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.+1890432\zeta(3)\zeta(5) +406080 M(2,6) -604288\zeta(9) +939840\zeta(3)\zeta(6) -304128\zeta(4)\zeta(5) \right. \nonumber \\ &\left.-152640\zeta(2)\zeta(7) +23040\zeta(3)^3 -579897\zeta(10) +519840\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-3240\zeta(3)^2\zeta(4) -185040\zeta(2)\zeta(3)\zeta(5) +203832\zeta(5)^2 +98280 M(2,8) \right. \nonumber \\ &\left.+11520\zeta(2) M(2,6)\right) \tag{1058} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{4}(k+1)^{3}} &= \frac{1}{144}\left( -257040\zeta(6) -32400\zeta(3)^2 +177534\zeta(7) +41040\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+47520\zeta(3)\zeta(4) +134527\zeta(8) +37440\zeta(2)\zeta(3)^2 -154368\zeta(3)\zeta(5) -33120 M(2,6) \right. \nonumber \\ &\left.+18884\zeta(9) -29370\zeta(3)\zeta(6) +9504\zeta(4)\zeta(5) +4770\zeta(2)\zeta(7) \right. \nonumber \\ &\left.-720\zeta(3)^3\right) \tag{1059} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{3}(k+1)^{4}} &= \frac{1}{144}\left( 257040\zeta(6) +32400\zeta(3)^2 -174006\zeta(7) -41040\zeta(2)\zeta(5) \right. \nonumber \\ &\left.-47520\zeta(3)\zeta(4) -132337\zeta(8) -36000\zeta(2)\zeta(3)^2 +148032\zeta(3)\zeta(5) +31680 M(2,6) \right. \nonumber \\ &\left.-14240\zeta(9) +25770\zeta(3)\zeta(6) -9504\zeta(4)\zeta(5) -4770\zeta(2)\zeta(7) \right. \nonumber \\ &\left.+720\zeta(3)^3\right) \tag{1060} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k^{2}(k+1)^{5}} &= \frac{1}{2304}\left( -2056320\zeta(6) -259200\zeta(3)^2 +1363824\zeta(7) \right. \nonumber \\ &\left.+328320\zeta(2)\zeta(5) +380160\zeta(3)\zeta(4) +1574904\zeta(8) +423360\zeta(2)\zeta(3)^2 \right. \nonumber \\ &\left.-1738368\zeta(3)\zeta(5) -371520 M(2,6) +455680\zeta(9) -824640\zeta(3)\zeta(6) +304128\zeta(4)\zeta(5) \right. \nonumber \\ &\left.+152640\zeta(2)\zeta(7) -23040\zeta(3)^3 +449109\zeta(10) -387360\zeta(3)\zeta(7) \right. \nonumber \\ &\left.-9720\zeta(3)^2\zeta(4) +124560\zeta(2)\zeta(3)\zeta(5) -122328\zeta(5)^2 -68040 M(2,8) \right. \nonumber \\ &\left.-11520\zeta(2) M(2,6)\right) \tag{1061} \\ \sum_{k=1}^\infty \frac{H(k)^{5}}{k(k+1)^{6}} &= \frac{-1}{2304}\left( -411264\zeta(6) -51840\zeta(3)^2 +267120\zeta(7) +65664\zeta(2)\zeta(5) \right. \nonumber \\ &\left.+76032\zeta(3)\zeta(4) +524968\zeta(8) +141120\zeta(2)\zeta(3)^2 -579456\zeta(3)\zeta(5) -123840 M(2,6) \right. \nonumber \\ &\left.+227840\zeta(9) -412320\zeta(3)\zeta(6) +152064\zeta(4)\zeta(5) +76320\zeta(2)\zeta(7) -11520\zeta(3)^3 \right. \nonumber \\ &\left.+449109\zeta(10) -387360\zeta(3)\zeta(7) -9720\zeta(3)^2\zeta(4) +124560\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.-122328\zeta(5)^2 -68040 M(2,8) -11520\zeta(2) M(2,6) -2668908\zeta(11) -275264\zeta(2)\zeta(9) \right. \nonumber \\ &\left.+993200\zeta(3)\zeta(8) +1403136\zeta(4)\zeta(7) +705120\zeta(5)\zeta(6) +65280\zeta(2)\zeta(3)^3 \right. \nonumber \\ &\left.-449280\zeta(3)^2\zeta(5) +34560\zeta(3) M(2,6) +92160 M(3,8)\right) \tag{1062} \end{align}\]

\[\begin{align} \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{5}(k+1)} &= \frac{1}{384}\left( 247296\zeta(7) +55680\zeta(2)\zeta(5) +114048\zeta(3)\zeta(4) \right. \nonumber \\ &\left.-280464\zeta(8) +15744\zeta(2)\zeta(3)^2 -187008\zeta(3)\zeta(5) -21888 M(2,6) +119584\zeta(9) \right. \nonumber \\ &\left.-209952\zeta(3)\zeta(6) +96768\zeta(4)\zeta(5) +31248\zeta(2)\zeta(7) -8704\zeta(3)^3 +814101\zeta(10) \right. \nonumber \\ &\left.-529680\zeta(3)\zeta(7) +253944\zeta(3)^2\zeta(4) +1200\zeta(2)\zeta(3)\zeta(5) -365064\zeta(5)^2 \right. \nonumber \\ &\left.-103128 M(2,8) -45120\zeta(2) M(2,6) -1469286\zeta(11) -166944\zeta(2)\zeta(9) +542488\zeta(3)\zeta(8) \right. \nonumber \\ &\left.+790176\zeta(4)\zeta(7) +410848\zeta(5)\zeta(6) +38720\zeta(2)\zeta(3)^3 -260352\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.+18240\zeta(3) M(2,6) +51200 M(3,8)\right) \tag{1063} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{4}(k+1)^{2}} &= \frac{1}{384}\left( -989184\zeta(7) -222720\zeta(2)\zeta(5) -456192\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+1113824\zeta(8) -62016\zeta(2)\zeta(3)^2 +742272\zeta(3)\zeta(5) +86592 M(2,6) -239168\zeta(9) \right. \nonumber \\ &\left.+419904\zeta(3)\zeta(6) -193536\zeta(4)\zeta(5) -62496\zeta(2)\zeta(7) +17408\zeta(3)^3 \right. \nonumber \\ &\left.-814101\zeta(10) +529680\zeta(3)\zeta(7) -253944\zeta(3)^2\zeta(4) -1200\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+365064\zeta(5)^2 +103128 M(2,8) +45120\zeta(2) M(2,6)\right) \tag{1064} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{3}(k+1)^{3}} &= \frac{1}{4}\left( 15456\zeta(7) +3480\zeta(2)\zeta(5) +7128\zeta(3)\zeta(4) -17278\zeta(8) \right. \nonumber \\ &\left.+954\zeta(2)\zeta(3)^2 -11508\zeta(3)\zeta(5) -1338 M(2,6) +2270\zeta(9) -4284\zeta(3)\zeta(6) \right. \nonumber \\ &\left.+1980\zeta(4)\zeta(5) +651\zeta(2)\zeta(7) -180\zeta(3)^3\right) \tag{1065} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k^{2}(k+1)^{4}} &= \frac{1}{384}\left( -989184\zeta(7) -222720\zeta(2)\zeta(5) -456192\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+1097760\zeta(8) -60096\zeta(2)\zeta(3)^2 +730752\zeta(3)\zeta(5) +84672 M(2,6) -196672\zeta(9) \right. \nonumber \\ &\left.+402624\zeta(3)\zeta(6) -186624\zeta(4)\zeta(5) -62496\zeta(2)\zeta(7) +17152\zeta(3)^3 \right. \nonumber \\ &\left.-779835\zeta(10) +490704\zeta(3)\zeta(7) -245544\zeta(3)^2\zeta(4) +15600\zeta(2)\zeta(3)\zeta(5) \right. \nonumber \\ &\left.+339864\zeta(5)^2 +94728 M(2,8) +45120\zeta(2) M(2,6)\right) \tag{1066} \\ \sum_{k=1}^\infty \frac{H(k)^{6}}{k(k+1)^{5}} &= \frac{-1}{384}\left( -247296\zeta(7) -55680\zeta(2)\zeta(5) -114048\zeta(3)\zeta(4) \right. \nonumber \\ &\left.+272432\zeta(8) -14784\zeta(2)\zeta(3)^2 +181248\zeta(3)\zeta(5) +20928 M(2,6) -98336\zeta(9) \right. \nonumber \\ &\left.+201312\zeta(3)\zeta(6) -93312\zeta(4)\zeta(5) -31248\zeta(2)\zeta(7) +8576\zeta(3)^3 -779835\zeta(10) \right. \nonumber \\ &\left.+490704\zeta(3)\zeta(7) -245544\zeta(3)^2\zeta(4) +15600\zeta(2)\zeta(3)\zeta(5) +339864\zeta(5)^2 \right. \nonumber \\ &\left.+94728 M(2,8) +45120\zeta(2) M(2,6) +1373598\zeta(11) +149024\zeta(2)\zeta(9) -524968\zeta(3)\zeta(8) \right. \nonumber \\ &\left.-724416\zeta(4)\zeta(7) -365168\zeta(5)\zeta(6) -36160\zeta(2)\zeta(3)^3 +240768\zeta(3)^2\zeta(5) \right. \nonumber \\ &\left.-16320\zeta(3) M(2,6) -46720 M(3,8)\right) \tag{1067} \end{align}\]

References↩︎

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1. The authors dedicate this paper to the memory of Jonathan and Peter Borwein, two giants of mathematical research who recently passed away. Jonathan in particular investigated Euler sums in some earlier studies that we reference.↩︎