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A263405 Expansion of Product_{k>=1} 1/(1-x^(3*k+1))^k.

%I A263405 #14 Nov 16 2024 17:32:11
%S A263405 1,0,0,0,1,0,0,2,1,0,3,2,1,4,6,2,6,10,6,8,20,14,13,30,29,22,50,54,43,
%T A263405 74,99,76,119,166,144,182,276,254,294,442,451,468,701,758,772,1088,
%U A263405 1270,1256,1698,2052,2067,2618,3294,3352,4065,5162,5430,6284,8050
%N A263405 Expansion of Product_{k>=1} 1/(1-x^(3*k+1))^k.
%H A263405 Vaclav Kotesovec, <a href="/A263405/b263405.txt">Table of n, a(n) for n = 0..5000</a>
%F A263405 G.f.: exp(Sum_{k>=1} x^(4*k)/(k*(1-x^(3*k))^2)).
%F A263405 a(n) ~ c * Zeta(3)^(19/108) * exp(-Pi^4/(3888*Zeta(3)) - Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(35/108) * 3^(23/27) * sqrt(Pi) * n^(73/108)), where c = 3^(1/6) * sqrt(2*Pi) * exp(A263031) / Gamma(1/3) = 1.107474840397395849254161220076423560365022...
%p A263405 with(numtheory):
%p A263405 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A263405       `if`(irem(d-1, 3)=0, (d-1)/3, 0),
%p A263405        d=divisors(j))*a(n-j), j=1..n)/n)
%p A263405     end:
%p A263405 seq(a(n), n=0..60);  # after _Alois P. Heinz_, Oct 17 2015
%t A263405 nmax = 60; CoefficientList[Series[Product[1/(1-x^(3*k+1))^k,{k,1,nmax}],{x,0,nmax}],x]
%t A263405 nmax = 60; CoefficientList[Series[E^Sum[x^(4*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A263405 Cf. A262876, A263031, A263406, A263414, A263415.
%K A263405 nonn
%O A263405 0,8
%A A263405 _Vaclav Kotesovec_, Oct 17 2015