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Sum of an infinite series involving generalized harmonic numbers of order 2.
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%I #10 Jul 14 2020 10:51:43

%S 1,0,8,8,8,5,7,1,4,0,9,6,1,6,6,7,0,5,7,2,3,2,3,4,3,1,7,1,4,7,0,2,7,1,

%T 5,2,8,3,9,4,2,7,9,8,7,3,2,1,4,7,8,2,3,8,7,6,4,0,8,0,5,3,2,5,0,9,0,3,

%U 9,3,0,9,5,0,9,7,5,0,6,5,2,1,9,5,4,0,5,5,3,5,5,1,2,5,3,4,4,4,2,8,9,5,8,6

%N Sum of an infinite series involving generalized harmonic numbers of order 2.

%H Kam Cheong Au, <a href="https://arxiv.org/abs/2007.03957">Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series</a>, arXiv:2007.03957 [math.NT], 2020. See p. 2.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Harmonic_number">Harmonic number</a>.

%F Sum_{n >= 2} (H(n-1,2) / n^3) * 2^n / Binomial(2n,n), where H(n-1,2) is the (n-1)th generalized harmonic number of order 2.

%F Equals Pi^3 * C/24 - Pi * Beta(4) - 3 * Pi^2 * Zeta(3)/128 + 527 * Zeta(5)/256 + Pi^4 * log(2)/384, where C is the Catalan constant and Beta the Dirichlet Beta function.

%e 0.108885714096166705723234317147027152839427987321478238764080532509...

%t Pi^3 Catalan/24 - Pi DirichletBeta[4] - 3Pi^2 Zeta[3] / 128 + 527 Zeta[5] / 256 + Pi^4 Log[2] / 384 // N[#, 104]& // RealDigits // First

%K nonn,cons

%O 0,3

%A _Jean-François Alcover_, Jul 12 2020