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A262946 Expansion of Product_{k>=1} 1/(1-x^(3*k-1))^(3*k-1).

%I A262946 #15 Apr 22 2021 08:31:24
%S A262946 1,0,2,0,3,5,4,10,13,15,37,31,61,87,99,178,228,286,477,552,816,1163,
%T A262946 1418,2077,2790,3507,5113,6478,8563,11888,15005,20100,27054,34055,
%U A262946 46002,59905,76436,102105,130879,168103,221954,281300,363743,472557,597579,772148
%N A262946 Expansion of Product_{k>=1} 1/(1-x^(3*k-1))^(3*k-1).
%C A262946 A262946(n)/A262947(n) ~ exp(3*(d1-d2)) * Gamma(1/3)^3 / (2*Pi)^(3/2), where d1 = A263030 and d2 = A263031. - _Vaclav Kotesovec_, Oct 08 2015
%H A262946 Vaclav Kotesovec, <a href="/A262946/b262946.txt">Table of n, a(n) for n = 0..5000</a>
%H A262946 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 A262946 Vaclav Kotesovec, <a href="/A262946/a262946.jpg">Graph - The asymptotic ratio (120000 terms)</a>
%F A262946 a(n) ~ (2*Zeta(3))^(5/36) * exp(3*d1 + (3/2)^(2/3) * Zeta(3)^(1/3) * n^(2/3)) / (3^(29/36) * Gamma(2/3) * n^(23/36)), where d1 = A263030 = Integral_{x=0..infinity} 1/x*(exp(-2*x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 1/(9*x) + exp(-x)/36) = -0.18870819197952853237641009864920797359211446726842922150941... . - _Vaclav Kotesovec_, Oct 08 2015
%p A262946 with(numtheory):
%p A262946 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A262946       `if`(irem(d+3, 3, 'r')=2, 3*r-1, 0),
%p A262946        d=divisors(j))*a(n-j), j=1..n)/n)
%p A262946     end:
%p A262946 seq(a(n), n=0..45);  # _Alois P. Heinz_, Oct 05 2015
%t A262946 nmax=60; CoefficientList[Series[Product[1/((1-x^(3k-1))^(3k-1)),{k,1,nmax}],{x,0,nmax}],x]
%t A262946 nmax=60; CoefficientList[Series[E^Sum[1/j*x^(2*j)*(2+x^(3*j))/(1-x^(3*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y A262946 Cf. A035386, A262811, A262876, A262883, A262923, A262947.
%K A262946 nonn
%O A262946 0,3
%A A262946 _Vaclav Kotesovec_, Oct 05 2015