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A263363 Expansion of Product_{k>=1} 1/(1-x^(k+7))^k.

%I A263363 #14 Nov 16 2024 16:10:27
%S A263363 1,0,0,0,0,0,0,0,1,2,3,4,5,6,7,8,10,12,17,22,32,42,59,76,103,130,171,
%T A263363 216,280,354,460,584,757,968,1249,1596,2056,2618,3354,4266,5441,6900,
%U A263363 8778,11108,14094,17814,22546,28450,35946,45280,57088,71806,90347
%N A263363 Expansion of Product_{k>=1} 1/(1-x^(k+7))^k.
%H A263363 Vaclav Kotesovec, <a href="/A263363/b263363.txt">Table of n, a(n) for n = 0..5000</a>
%H A263363 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 A263363 G.f.: exp(Sum_{k>=1} x^(8*k)/(k*(1-x^k)^2)).
%F A263363 a(n) ~ exp(1/12 - 49*Pi^4/(432*Zeta(3)) - 7*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^(269/36) * Pi^3 / (398131200 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(287/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
%p A263363 with(numtheory):
%p A263363 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263363       max(0, d-7), d=divisors(j))*a(n-j), j=1..n)/n)
%p A263363     end:
%p A263363 seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 16 2015
%t A263363 nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+7))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263363 nmax = 60; CoefficientList[Series[E^Sum[x^(8*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A263363 Cf. A000219, A052847, A263358, A263359, A263360, A263361, A263362, A263364.
%K A263363 nonn
%O A263363 0,10
%A A263363 _Vaclav Kotesovec_, Oct 16 2015