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Expansion of Product_{k>=1} 1/(1 - x^(2*k+7))^k.
5

%I #15 Nov 16 2024 16:33:22

%S 1,0,0,0,0,0,0,0,0,1,0,2,0,3,0,4,0,5,1,6,2,7,6,8,10,9,19,11,28,13,44,

%T 18,60,27,85,42,111,67,148,109,188,169,245,260,313,390,408,568,535,

%U 811,717,1139,974,1568,1343,2134,1872,2873,2621,3832,3687,5088

%N Expansion of Product_{k>=1} 1/(1 - x^(2*k+7))^k.

%H Vaclav Kotesovec, <a href="/A263396/b263396.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^(9*k)/(k*(1-x^(2*k))^2)).

%F a(n) ~ 8 * 2^(1/72) * exp(-1/24 - 49*Pi^4/(1728*Zeta(3)) - 7*Pi^2 * n^(1/3) / (12 * 2^(2/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * sqrt(A) * n^(97/72) / (45*sqrt(3*Pi) * Zeta(3)^(133/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

%p with(numtheory):

%p a:= proc(n) option remember; local r; `if`(n=0, 1,

%p add(add(`if`(irem(d-6, 2, 'r')=1, d*r, 0)

%p , d=divisors(j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..65); # _Alois P. Heinz_, Oct 17 2015

%t nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+7))^k, {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 60; CoefficientList[Series[E^Sum[x^(9*k)/(k*(1-x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A035528, A263150, A263352, A263395, A263397.

%K nonn,changed

%O 0,12

%A _Vaclav Kotesovec_, Oct 16 2015