A255702 Decimal expansion of the Plouffe sum S(5,2) = Sum_{n >= 1} 1/(n^5*(exp(2*Pi*n)-1)).

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

Offset

-2,2

Links

Table of n, a(n) for n=-2..101.

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=2 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(2*Pi*k)). - Amiram Eldar, Jun 05 2023

Example

0.0018710456053012595148739514758105634303189628230875828656...

Mathematica

digits = 104; S[5, 2] = NSum[1/(n^5*(Exp[2*Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[5, 2], 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)), A255703 (S(5,4)).

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

Sequence in context: A259348 A257238 A348728 * A154401 A248582 A362120

Adjacent sequences: A255699 A255700 A255701 * A255703 A255704 A255705

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Mar 02 2015

Status

approved