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A258346 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)*(k-2)/6).

%I A258346 #6 May 28 2015 03:37:37
%S A258346 1,0,0,1,4,10,20,39,72,144,280,567,1112,2187,4204,8073,15309,28986,
%T A258346 54548,102286,190881,354717,656194,1208712,2217624,4052633,7379630,
%U A258346 13390098,24215587,43649482,78435884,140513905,250988186,447037367,794031641,1406585604
%N A258346 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)*(k-2)/6).
%H A258346 Vaclav Kotesovec, <a href="/A258346/b258346.txt">Table of n, a(n) for n = 0..1000</a>
%F A258346 a(n) ~ (3*Zeta(5))^(1/10) / (2^(523/720) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (10497600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (16200000 * Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + (-343*Pi^12 / (2430000000 * 2^(3/5) * 15^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (1080000 * 2^(1/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * (15*Zeta(5))^(2/5))) * n^(2/5) - 7*Pi^4 / (180 * 2^(4/5) * (15*Zeta(5))^(3/5)) * n^(3/5) + 5*(15*Zeta(5))^(1/5) / 2^(12/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.
%t A258346 nmax=50; CoefficientList[Series[Product[(1+x^k)^(k*(k-1)*(k-2)/6),{k,1,nmax}],{x,0,nmax}],x]
%Y A258346 Cf. A248882, A028377, A258341, A258342, A258343, A258344, A258345, A258352.
%K A258346 nonn
%O A258346 0,5
%A A258346 _Vaclav Kotesovec_, May 27 2015