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%I #27 Jun 05 2023 02:20:50
%S 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,
%T 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,
%U 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
%N Decimal expansion of the Plouffe sum S(1,1) = Sum_{n >= 1} 1/(n*(exp(Pi*n)-1)).
%H Steven R. Finch, <a href="https://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020-2022, p. 6.
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/inspired2.pdf">Identities inspired by Ramanujan Notebooks (part 2)</a>, April 2006.
%H Linas Vepštas, <a href="https://doi.org/10.1007/s11139-011-9335-9">On Plouffe's Ramanujan identities</a>, The Ramanujan Journal, Vol. 27 (2012), pp. 387-408; <a href="https://cyberleninka.org/article/n/534457.pdf">alternative link</a>; <a href="https://arxiv.org/abs/math/0609775">arXiv preprint</a>, arXiv:math/0609775 [math.NT], 2006-2010.
%F This is the case k = m = 1 of S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).
%F Pi = 72*S(1,1) - 96*S(1,2) + 24*S(1,4).
%F Equals Sum_{k>=1} sigma(k)/(k*exp(Pi*k)). - _Amiram Eldar_, Jun 05 2023
%e 0.046128978779350100707761371934335281367940770498217...
%t 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
%Y 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)).
%Y Cf. A000203 (sigma), A000796 (Pi).
%K nonn,cons,easy
%O -1,1
%A _Jean-François Alcover_, Mar 02 2015
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