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A263397 Expansion of Product_{k>=1} 1/(1 - x^(2*k+9))^k.

%I A263397 #19 Feb 16 2025 08:33:27
%S A263397 1,0,0,0,0,0,0,0,0,0,0,1,0,2,0,3,0,4,0,5,0,6,1,7,2,8,6,9,10,10,19,11,
%T A263397 28,13,44,15,60,20,85,29,110,44,146,69,183,111,233,171,286,262,358,
%U A263397 391,441,568,553,808,697,1129,898,1543,1174,2080,1563,2766
%N A263397 Expansion of Product_{k>=1} 1/(1 - x^(2*k+9))^k.
%C A263397 In general, if g.f. = Product_{k>=1} 1/(1-x^(2*k+v))^k and v>0 is odd, then a(n) ~ d2(v) * (2*n)^(v^2/24 - 25/36) * exp(-Pi^4 * v^2 / (1728*Zeta(3)) - Pi^2 * v * n^(1/3) /(3 * 2^(8/3) * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) / (sqrt(3*Pi) * Zeta(3)^(v^2/24 - 7/36)), where Zeta(3) = A002117.
%C A263397 d2(v) = exp(Integral_{x=0..infinity} 1/(x*exp((v-2)*x) * (exp(2*x)- 1)^2) - (3*v^2-2)/(24*x*exp(x)) + v/(4*x^2) - 1/(4*x^3) dx).
%C A263397 d2(v) = 2^(v/4 - 1/12) * exp(-Zeta'(-1)/2) / Product_{j=1..(v-1)/2} (2*j-1)!!, where Zeta'(-1) = A084448 and Product_{j=1..(v-1)/2} (2*j-1)!! = A057863((v-1)/2).
%C A263397 d2(v) = 2^(1/12 + v/4 - v^2/8) * exp(1/12) * Pi^(v/4) / (A * G(v/2 + 1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.
%H A263397 Vaclav Kotesovec, <a href="/A263397/b263397.txt">Table of n, a(n) for n = 0..5000</a>
%H A263397 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
%H A263397 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.
%F A263397 G.f.: exp(Sum_{k>=1} x^(11*k)/(k*(1-x^(2*k))^2)).
%F A263397 a(n) ~ 16 * 2^(61/72) * exp(-1/24 - 3*Pi^4/(64*Zeta(3)) - 3*Pi^2 * n^(1/3) / (2^(8/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * sqrt(A) * n^(193/72) / (4725*sqrt(3*Pi) * Zeta(3)^(229/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
%p A263397 with(numtheory):
%p A263397 a:= proc(n) option remember; local r; `if`(n=0, 1,
%p A263397        add(add(`if`(irem(d-8, 2, 'r')=1, d*r, 0)
%p A263397        , d=divisors(j))*a(n-j), j=1..n)/n)
%p A263397     end:
%p A263397 seq(a(n), n=0..65);  # _Alois P. Heinz_, Oct 17 2015
%t A263397 nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+9))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263397 nmax = 60; CoefficientList[Series[E^Sum[x^(11*k)/(k*(1-x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A263397 Cf. A035528 (v=-1), A263150 (v=1), A263352 (v=3), A263395 (v=5), A263396 (v=7).
%K A263397 nonn
%O A263397 0,14
%A A263397 _Vaclav Kotesovec_, Oct 16 2015