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A263358 Expansion of Product_{k>=1} 1/(1-x^(k+2))^k.

%I A263358 #21 Feb 16 2025 08:33:27
%S A263358 1,0,0,1,2,3,5,7,12,18,29,43,69,101,155,231,347,509,759,1106,1626,
%T A263358 2359,3428,4938,7127,10194,14587,20756,29498,41716,58932,82888,116413,
%U A263358 162924,227602,316988,440696,610953,845469,1167118,1608178,2210888,3034124,4155111
%N A263358 Expansion of Product_{k>=1} 1/(1-x^(k+2))^k.
%C A263358 In general, if v>=0 and g.f. = Product_{k>=1} 1/(1-x^(k+v))^k, then a(n) ~ d1(v) * n^(v^2/6 - 25/36) * exp(-Pi^4 * v^2 / (432*Zeta(3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(2/3) - v * Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * 2^(v^2/6 + 11/36) * Zeta(3)^(v^2/6 - 7/36)), where Zeta(3) = A002117.
%C A263358 d1(v) = exp(Integral_{x=0..infinity} (1/(x*exp((v-1)*x) * (exp(x)-1)^2) - (6*v^2-1) / (12*x*exp(x)) + v/x^2 - 1/x^3) dx).
%C A263358 d1(v) = (exp(Zeta'(-1) - v*Zeta'(0))) / Product_{j=0..v-1} j!, where Zeta'(0) = -A075700, Zeta'(-1) = A084448 and Product_{j=0..v-1} j! = A000178(v-1).
%C A263358 d1(v) = exp(1/12) * (2*Pi)^(v/2) / (A * G(v+1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.
%H A263358 Vaclav Kotesovec, <a href="/A263358/b263358.txt">Table of n, a(n) for n = 0..5000</a>
%H A263358 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 A263358 Vaclav Kotesovec, <a href="/A263358/a263358.jpg">Graph - The asymptotic ratio (30000 terms)</a>
%H A263358 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>
%F A263358 G.f.: exp(Sum_{k>=1} x^(3*k)/(k*(1-x^k)^2)).
%F A263358 a(n) ~ exp(1/12 - Pi^4/(108*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(1/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(1/36) * sqrt(Pi) / (A * sqrt(3) * Zeta(3)^(17/36) * n^(1/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
%p A263358 with(numtheory):
%p A263358 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263358       max(0, d-2), d=divisors(j))*a(n-j), j=1..n)/n)
%p A263358     end:
%p A263358 seq(a(n), n=0..50);  # _Alois P. Heinz_, Oct 16 2015
%t A263358 nmax = 50; CoefficientList[Series[Product[1/(1-x^(k+2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263358 nmax = 50; CoefficientList[Series[E^Sum[x^(3*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A263358 Cf. A000219, A052847, A263359, A263360, A263361, A263362, A263363, A263364.
%K A263358 nonn
%O A263358 0,5
%A A263358 _Vaclav Kotesovec_, Oct 16 2015