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A263150 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^k.

%I A263150 #17 Oct 17 2015 07:55:06
%S A263150 1,0,0,1,0,2,1,3,2,5,6,7,11,12,21,22,34,38,59,67,95,118,155,198,252,
%T A263150 330,409,540,662,867,1067,1382,1705,2187,2705,3430,4267,5348,6666,
%U A263150 8303,10352,12812,15964,19681,24467,30091,37282,45769,56539,69296,85304
%N A263150 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^k.
%C A263150 The side effect of this calculation is a formula: Integral_{x=0..infinity} exp(-3*x)/(x*(1-exp(-2*x))^2) - 1/(4*x^3) + 1/(4*x^2) - exp(-x)/(24*x) = log(2)/6 + log(A)/2 - 1/24, where A = A074962 is the Glaisher-Kinkelin constant.
%H A263150 Vaclav Kotesovec, <a href="/A263150/b263150.txt">Table of n, a(n) for n = 0..10000</a>
%H A263150 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015
%F A263150 G.f.: exp(Sum_{j>=1} 1/j*x^(3*j)/(1 - x^(2*j))^2).
%F A263150 a(n) ~ sqrt(A) * Zeta(3)^(11/72) * exp(-1/24 - Pi^4/(1728*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(8/3)* Zeta(3)^(1/3)) + 3 * (Zeta(3)/2)^(1/3) * n^(2/3)/2) / (2^(35/72) * sqrt(3*Pi) * n^(47/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
%p A263150 with(numtheory):
%p A263150 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263150       `if`(irem(d-1, 2)=0, (d-1)/2, 0),
%p A263150        d=divisors(j))*a(n-j), j=1..n)/n)
%p A263150     end:
%p A263150 seq(a(n), n=0..60);  # after _Alois P. Heinz_, Oct 17 2015
%t A263150 nmax = 100; CoefficientList[Series[Product[1/(1-x^(2*k+1))^k,{k,1,nmax}],{x,0,nmax}],x]
%t A263150 nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y A263150 Cf. A035528, A052847, A263140, A263149, A263199, A263352, A263395, A263396, A263397.
%K A263150 nonn
%O A263150 0,6
%A A263150 _Vaclav Kotesovec_, Oct 10 2015