A255695 Decimal expansion of the Plouffe sum S(1,1) = Sum_{n >= 1} 1/(n*(exp(Pi*n)-1)).

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

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 = m = 1 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(Pi*k)). - Amiram Eldar, Jun 05 2023

Example

0.046128978779350100707761371934335281367940770498217...

Mathematica

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

Crossrefs

Cf. 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)), A255703 (S(5,4)).

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

Sequence in context: A248938 A106144 A154478 * A246489 A343624 A309445

Adjacent sequences: A255692 A255693 A255694 * A255696 A255697 A255698

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Mar 02 2015

Status

approved