A255700 Decimal expansion of the Plouffe sum S(3,4) = Sum_{n >= 1} 1/(n^3*(exp(4*Pi*n)-1)).

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

Offset

-5,1

Links

Table of n, a(n) for n=-5..98.

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=3, m=4 of S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).

Pi^3 = 720*S(3,1) - 900*S(3,2) + 180*S(3,4).

zeta(3) = 28*S(3,1) - 37*S(3,2) + 7*S(3,4).

Equals Sum_{k>=1} sigma_3(k)/(k^3*exp(4*Pi*k)). - Amiram Eldar, Jun 05 2023

Example

0.000003487356038004276054514730322548976264651146827033884525679...

Mathematica

digits = 104; S[3, 4] = NSum[1/(n^3*(Exp[4*Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[3, 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)), A255701 (S(5,1)), A255702 (S(5,2)), A255703 (S(5,4)).

Cf. A001158 (sigma_3), A002117 (zeta(3)).

Sequence in context: A105753 A292822 A255703 * A255697 A019972 A064406

Adjacent sequences: A255697 A255698 A255699 * A255701 A255702 A255703

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Mar 02 2015

Status

approved