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Expansion of Product_{k>=1} 1/(1-x^(2*k-1))^(2*k-1).
18

%I #17 Sep 08 2017 06:50:24

%S 1,1,1,4,4,9,15,22,37,56,92,133,210,310,466,696,1013,1495,2160,3141,

%T 4495,6462,9172,13024,18387,25840,36213,50500,70280,97302,134522,

%U 185105,254245,347938,475036,646676,878145,1189468,1607095,2166672,2913794,3910741

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

%H Alois P. Heinz, <a href="/A262811/b262811.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 a(n) ~ exp(-1/12 + 3*Zeta(3)^(1/3)*n^(2/3)/2) * A * Zeta(3)^(5/36) / (2^(2/3) * sqrt(3*Pi) * n^(23/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

%F a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A050999(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 09 2017

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*

%p `if`(d::even, 0, d), d=divisors(j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..45); # _Alois P. Heinz_, Oct 05 2015

%t nmax = 60; CoefficientList[Series[Product[1/(1-x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A000219, A003293, A035528, A161870, A262736, A292038.

%K nonn

%O 0,4

%A _Vaclav Kotesovec_, Oct 03 2015