A255701 Decimal expansion of the Plouffe sum S(5,1) = Sum_{n >= 1} 1/(n^5*(exp(Pi*n)-1)).

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

Offset

-1,1

Links

Table of n, a(n) for n=-1..102.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022, p. 6.

Simon Plouffe, Identities inspired by Ramanujan Notebooks (part 2), April 2006.

Linas Vepštas, On Plouffe's Ramanujan identities, The Ramanujan Journal, Vol. 27 (2012), pp. 387-408; alternative link; arXiv preprint, arXiv:math/0609775 [math.NT], 2006-2010.

Formula

This is the case k=5, m=1 of S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).

zeta(5) = 24*S(5,1) - (259/10)*S(5,2) - (1/10)*S(5,4).

Equals Sum_{k>=1} sigma_5(k)/(k^5*exp(Pi*k)). - Amiram Eldar, Jun 05 2023

Example

0.0452245077106734305608511495517055571453316321950147201921...

Mathematica

digits = 104; S[5, 1] = NSum[1/(n^5*(Exp[Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[5, 1], 10, digits] // First

Crossrefs

Cf. A255695 (S(1,1)), A084254 (S(1,2)), A255697 (S(1,4)), A255698 (S(3,1)), A255699 (S(3,2)), A255700 (S(3,4)), A255702 (S(5,2)), A255703 (S(5,4)).

Cf. A001160 (sigma_5), A013663 (zeta(5)).

Sequence in context: A267095 A016715 A337192 * A085548 A329957 A074459

Adjacent sequences: A255698 A255699 A255700 * A255702 A255703 A255704

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Mar 02 2015

Status

approved