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A262947 Expansion of Product_{k>=1} 1/(1-x^(3*k-2))^(3*k-2).

%I A262947 #18 Apr 22 2021 08:30:55
%S A262947 1,1,1,1,5,5,5,12,22,22,32,60,80,93,161,231,282,404,616,775,1041,1535,
%T A262947 2037,2600,3708,5029,6411,8710,11968,15315,20189,27444,35619,45939,
%U A262947 61605,80422,102932,135481,177391,226263,293561,382984,488826,626558,812750
%N A262947 Expansion of Product_{k>=1} 1/(1-x^(3*k-2))^(3*k-2).
%C A262947 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 A262947 Vaclav Kotesovec, <a href="/A262947/b262947.txt">Table of n, a(n) for n = 0..5000</a>
%H A262947 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 A262947 Vaclav Kotesovec, <a href="/A262947/a262947.jpg">Graph - The asymptotic ratio (120000 terms)</a>
%F A262947 a(n) ~ 2^(23/36) * sqrt(Pi) * Zeta(3)^(5/36) * exp(3*d2 + (3/2)^(2/3) * Zeta(3)^(1/3) * n^(2/3)) / (3^(11/36) * Gamma(1/3)^2 * n^(23/36)), where d2 = A263031 = Integral_{x=0..infinity} 1/x*(exp(-x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 2/(9*x) - 5*exp(-x)/36) = -0.01453742918328403360502029450226209036054149... . - _Vaclav Kotesovec_, Oct 08 2015
%p A262947 with(numtheory):
%p A262947 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A262947       `if`(irem(d+3, 3, 'r')=1, 3*r-2, 0),
%p A262947        d=divisors(j))*a(n-j), j=1..n)/n)
%p A262947     end:
%p A262947 seq(a(n), n=0..45);  # _Alois P. Heinz_, Oct 05 2015
%t A262947 nmax=60; CoefficientList[Series[Product[1/((1-x^(3k-2))^(3k-2)),{k,1,nmax}],{x,0,nmax}],x]
%t A262947 nmax=60; CoefficientList[Series[E^Sum[1/j*x^j*(1+2*x^(3*j))/(1-x^(3*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y A262947 Cf. A035382, A262877, A262883, A262923, A262946.
%K A262947 nonn
%O A262947 0,5
%A A262947 _Vaclav Kotesovec_, Oct 05 2015