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%I #14 Nov 16 2024 16:10:35
%S 1,0,0,0,0,0,1,2,3,4,5,6,8,10,15,20,30,40,58,76,106,140,191,252,344,
%T 454,613,814,1091,1442,1926,2538,3368,4432,5852,7678,10107,13222,
%U 17337,22636,29582,38518,50195,65198,84712,109784,142254,183924,237742,306688
%N Expansion of Product_{k>=1} 1/(1-x^(k+5))^k.
%H Vaclav Kotesovec, <a href="/A263361/b263361.txt">Table of n, a(n) for n = 0..5000</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 G.f.: exp(Sum_{k>=1} x^(6*k)/(k*(1-x^k)^2)).
%F a(n) ~ exp(1/12 - 25*Pi^4/(432*Zeta(3)) - 5*Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(125/36) * Pi^2 / (576 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(143/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
%p with(numtheory):
%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p max(0, d-5), d=divisors(j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Oct 16 2015
%t nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+5))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 60; CoefficientList[Series[E^Sum[x^(6*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A000219, A052847, A263358, A263359, A263360, A263362, A263363, A263364.
%K nonn
%O 0,8
%A _Vaclav Kotesovec_, Oct 16 2015