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A263406 Expansion of Product_{k>=1} 1/(1-x^(3*k+2))^k.

%I A263406 #13 Nov 16 2024 17:31:50
%S A263406 1,0,0,0,0,1,0,0,2,0,1,3,0,2,4,1,6,5,2,10,7,6,19,9,14,29,14,28,46,23,
%T A263406 53,66,43,95,99,76,158,143,141,256,217,247,403,326,432,617,509,720,
%U A263406 935,801,1187,1399,1281,1892,2087,2047,2983,3107,3272,4589,4647
%N A263406 Expansion of Product_{k>=1} 1/(1-x^(3*k+2))^k.
%H A263406 Vaclav Kotesovec, <a href="/A263406/b263406.txt">Table of n, a(n) for n = 0..5000</a>
%F A263406 G.f.: exp(Sum_{k>=1} x^(5*k)/(k*(1-x^(3*k))^2)).
%F A263406 a(n) ~ c * Zeta(3)^(13/108) * exp(-Pi^4/(972*Zeta(3)) - Pi^2 * n^(1/3) / (2^(1/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(41/108) * 3^(20/27) * sqrt(Pi) * n^(67/108)), where c = 3^(1/3) * Gamma(1/3) * exp(A263030) / sqrt(2*Pi) = 1.2763162741536982965216627321306598385267089489...
%p A263406 with(numtheory):
%p A263406 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263406       `if`(irem(d-2, 3)=0, (d-2)/3, 0),
%p A263406        d=divisors(j))*a(n-j), j=1..n)/n)
%p A263406     end:
%p A263406 seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 17 2015
%t A263406 nmax = 60; CoefficientList[Series[Product[1/(1-x^(3*k+2))^k,{k,1,nmax}],{x,0,nmax}],x]
%t A263406 nmax = 60; CoefficientList[Series[E^Sum[x^(5*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A263406 Cf. A262877, A263030, A263405, A263414, A263415.
%K A263406 nonn
%O A263406 0,9
%A A263406 _Vaclav Kotesovec_, Oct 17 2015