%I #30 Feb 16 2025 22:35:44
%S 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,
%T 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,
%U 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
%N Decimal expansion of the Plouffe sum S(1,4) = Sum_{n >= 1} 1/(n*(exp(4*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://arxiv.org/abs/math/0609775">arXiv preprint</a>, arXiv:math/0609775 [math.NT], 2006-2010.
%F This is the case k=1, m=4 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(4*Pi*k)). - _Amiram Eldar_, Jun 05 2023
%e 0.000003487360598600608547009753485704881044662756456465722...
%t 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
%Y 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)).
%Y Cf. A000203 (sigma), A000796 (Pi).
%K nonn,cons,easy,changed
%O -5,1
%A _Jean-François Alcover_, Mar 02 2015