A255697 Decimal expansion of the Plouffe sum S(1,4) = Sum_{n >= 1} 1/(n*(exp(4*Pi*n)-1)).

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

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

Pi = 72*S(1,1) - 96*S(1,2) + 24*S(1,4).

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

Example

0.000003487360598600608547009753485704881044662756456465722...

Mathematica

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

Crossrefs

Cf. A255695 (S(1,1)), A084254 (S(1,2)), A255698 (S(3,1)), A255699 (S(3,2)), A255700 (S(3,4)), A255701 (S(5,1)), A255702 (S(5,2)), A255703 (S(5,4)).

Cf. A000203 (sigma), A000796 (Pi).

Sequence in context: A292822 A255703 A255700 * A019972 A064406 A299068

Adjacent sequences: A255694 A255695 A255696 * A255698 A255699 A255700

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Mar 02 2015

Status

approved