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%I #12 Nov 11 2020 08:49:52
%S 1,0,1,3,7,13,28,52,107,203,396,741,1409,2596,4813,8777,15972,28737,
%T 51553,91644,162288,285377,499653,869758,1507615,2599974,4465606,
%U 7635607,13005252,22061424,37287395,62788012,105365891,176211393,293741195,488101711,808604106
%N Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)/2).
%H Vaclav Kotesovec, <a href="/A258349/b258349.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) ~ 1 / (2^(155/96) * 15^(11/96) * Pi^(1/24) * n^(59/96)) * exp(-Zeta'(-1)/2 - Zeta(3) / (8*Pi^2) - 75*Zeta(3)^3 / (2*Pi^8) - 15^(5/4) * Zeta(3)^2 / (2^(7/4) * Pi^5) * n^(1/4) - sqrt(15/2) * Zeta(3) / Pi^2 * sqrt(n) + 2^(7/4)*Pi / (3*15^(1/4)) * n^(3/4)), where Zeta(3) = A002117, Zeta'(-1) = A084448 = 1/12 - log(A074962).
%F G.f.: exp(Sum_{k>=1} (sigma_3(k) - sigma_2(k))*x^k/(2*k)). - _Ilya Gutkovskiy_, Aug 22 2018
%t nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)/2),{k,1,nmax}],{x,0,nmax}],x]
%o (SageMath) # uses[EulerTransform from A166861]
%o b = EulerTransform(lambda n: binomial(n,2))
%o print([b(n) for n in range(37)]) # _Peter Luschny_, Nov 11 2020
%Y Cf. A000294, A023871, A027999, A258347, A258348, A258350, A258351, A258352.
%K nonn
%O 0,4
%A _Vaclav Kotesovec_, May 27 2015
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