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A263352 Expansion of Product_{k>=1} 1/(1 - x^(2*k+3))^k.

%I A263352 #18 Feb 16 2025 08:33:27
%S A263352 1,0,0,0,0,1,0,2,0,3,1,4,2,5,6,7,10,9,19,14,29,23,46,38,66,64,99,107,
%T A263352 143,171,211,270,311,418,465,633,698,945,1049,1399,1579,2052,2364,
%U A263352 2997,3527,4366,5219,6339,7686,9197,11234,13321,16340,19261,23622,27796
%N A263352 Expansion of Product_{k>=1} 1/(1 - x^(2*k+3))^k.
%C A263352 From _Vaclav Kotesovec_, Oct 17 2015: (Start)
%C A263352 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 A263352 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 A263352 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 A263352 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.
%C A263352 (End)
%H A263352 Vaclav Kotesovec, <a href="/A263352/b263352.txt">Table of n, a(n) for n = 0..1000</a>
%H A263352 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 A263352 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.
%F A263352 a(n) ~ 2^(25/72) * sqrt(A) * exp(-1/24 + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3) - Pi^4/(192*Zeta(3)) - Pi^2 * n^(1/3)/(2^(8/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * Zeta(3)^(13/72) * n^(23/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
%p A263352 with(numtheory):
%p A263352 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263352       `if`(d>1 and d::odd, (d-3)/2, 0),
%p A263352       d=divisors(j))*a(n-j), j=1..n)/n)
%p A263352     end:
%p A263352 seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 17 2015
%t A263352 nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%Y A263352 Cf. A052847, A000219, A263150, A035528, A263395, A263396, A263397.
%K A263352 nonn
%O A263352 0,8
%A A263352 _Vaclav Kotesovec_, Oct 16 2015