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A263359 Expansion of Product_{k>=1} 1/(1-x^(k+3))^k.

%I A263359 #14 Nov 16 2024 16:10:46
%S A263359 1,0,0,0,1,2,3,4,6,8,13,18,29,40,61,86,127,178,260,364,524,734,1042,
%T A263359 1454,2051,2848,3981,5510,7652,10542,14558,19970,27428,37480,51222,
%U A263359 69720,94870,128634,174306,235506,317899,428018,575688,772540,1035538,1385264
%N A263359 Expansion of Product_{k>=1} 1/(1-x^(k+3))^k.
%H A263359 Vaclav Kotesovec, <a href="/A263359/b263359.txt">Table of n, a(n) for n = 0..5000</a>
%H A263359 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 A263359 G.f.: exp(Sum_{k>=1} x^(4*k)/(k*(1-x^k)^2)).
%F A263359 a(n) ~ exp(1/12 - Pi^4/(48*Zeta(3)) - Pi^2 * n^(1/3) / (2^(4/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(29/36) * Pi / (A * 2^(47/36) * sqrt(3) * Zeta(3)^(47/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
%p A263359 with(numtheory):
%p A263359 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263359       max(0, d-3), d=divisors(j))*a(n-j), j=1..n)/n)
%p A263359     end:
%p A263359 seq(a(n), n=0..50);  # _Alois P. Heinz_, Oct 16 2015
%t A263359 nmax = 50; CoefficientList[Series[Product[1/(1-x^(k+3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263359 nmax = 50; CoefficientList[Series[E^Sum[x^(4*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A263359 Cf. A000219, A052847, A263358, A263360, A263361, A263362, A263363, A263364.
%K A263359 nonn
%O A263359 0,6
%A A263359 _Vaclav Kotesovec_, Oct 16 2015