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%I #10 Dec 01 2020 17:07:02
%S 1,2,4,9,19,37,72,134,246,442,782,1359,2338,3964,6652,11046,18176,
%T 29631,47935,76931,122608,194072,305269,477258,741977,1147227,1764778,
%U 2701403,4115892,6242846,9428575,14181272,21245738,31708402,47150928,69867001,103176007,151864745,222821779
%N Expansion of (1 + x) * Product_{k>=1} 1/(1 - x^k)^k.
%H Robert Israel, <a href="/A309267/b309267.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = A000219(n) + A000219(n-1).
%F a(n) ~ Zeta(3)^(7/36) * 2^(25/36) * exp(3 * Zeta(3)^(1/3) * (n/2)^(2/3) + 1/12) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Jul 20 2019
%p G:= (1+x)/mul((1-x^k)^k,k=1..100):
%p S:= series(G,x,101):
%p seq(coeff(S,x,j),j=0..100); # _Robert Israel_, Dec 01 2020
%t nmax = 38; CoefficientList[Series[(1 + x) Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[2, k] a[n - k], {k, 1, n}]/n; Table[a[n] + a[n - 1], {n, 0, 38}]
%Y Cf. A000219, A052816, A084376, A091360, A191659, A277963.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jul 20 2019
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