A255703 Decimal expansion of the Plouffe sum S(5,4) = Sum_{n >= 1} 1/(n^5*(exp(4*Pi*n)-1)).

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

Offset

-5,1

Links

Table of n, a(n) for n=-5..96.

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=4 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(4*Pi*k)). - Amiram Eldar, Jun 05 2023

Example

0.000003487354897856938267595604056105856171160604720764172...

Mathematica

digits = 102; S[5, 4] = NSum[1/(n^5*(Exp[4*Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[5, 4], 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)), A255701 (S(5,1)), A255702 (S(5,2)).

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

Sequence in context: A328876 A105753 A292822 * A255700 A255697 A019972

Adjacent sequences: A255700 A255701 A255702 * A255704 A255705 A255706

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Mar 02 2015

Status

approved