A336211 Sum of an infinite series involving generalized harmonic numbers of order 2.

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, 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, 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

Offset

0,3

Links

Table of n, a(n) for n=0..103.

Kam Cheong Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, arXiv:2007.03957 [math.NT], 2020. See p. 2.

Eric Weisstein's World of Mathematics, Harmonic Number.

Wikipedia, Harmonic number.

Formula

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.

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.

Example

0.108885714096166705723234317147027152839427987321478238764080532509...

Mathematica

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

Crossrefs

Sequence in context: A023411 A181122 A023412 * A343392 A343393 A023413

Adjacent sequences: A336208 A336209 A336210 * A336212 A336213 A336214

Keyword

nonn,cons

Author

Jean-François Alcover, Jul 12 2020

Status

approved