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A263199 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^(2*k+1).

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%I A263199 #10 Oct 12 2015 10:39:39
%S A263199 1,0,0,3,0,5,6,7,15,19,36,41,77,100,156,230,317,482,665,981,1354,1967,
%T A263199 2710,3852,5363,7453,10373,14287,19780,27022,37220,50583,69140,93693,
%U A263199 127098,171640,231469,311323,417627,559577,747122,996947,1325872,1761900
%N A263199 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^(2*k+1).
%H A263199 Vaclav Kotesovec, <a href="/A263199/b263199.txt">Table of n, a(n) for n = 0..10000</a>
%H A263199 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 A263199 For n>1, a(n) = A262811(n) - A262811(n-1).
%F A263199 a(n) ~ A * Zeta(3)^(17/36) * exp(-1/12 + 3 * Zeta(3)^(1/3) * n^(2/3)/2) / (2^(2/3) * sqrt(3*Pi) * n^(35/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
%p A263199 with(numtheory):
%p A263199 b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263199       `if`(d::even, 0, d), d=divisors(j))*b(n-j), j=1..n)/n)
%p A263199     end:
%p A263199 seq(b(n)-b(n-1), n=0..60);  # after _Alois P. Heinz_
%t A263199 nmax = 100; CoefficientList[Series[Product[1/(1 - x^(2*k+1))^(2*k+1),{k,1,nmax}],{x,0,nmax}],x]
%Y A263199 Cf. A000219, A035528, A262811, A263140, A263149, A263150.
%K A263199 nonn
%O A263199 0,4
%A A263199 _Vaclav Kotesovec_, Oct 12 2015