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A258345 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)*(k-2)).

%I A258345 #6 May 28 2015 03:37:48
%S A258345 1,0,0,6,24,60,135,354,972,2684,6990,17802,44627,111582,277329,684164,
%T A258345 1671984,4050096,9735209,23238480,55120950,129940442,304502583,
%U A258345 709464798,1643920584,3789158988,8690016942,19833550266,45056952957,101900481462,229462378987
%N A258345 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)*(k-2)).
%H A258345 Vaclav Kotesovec, <a href="/A258345/b258345.txt">Table of n, a(n) for n = 0..1000</a>
%F A258345 a(n) ~ 3^(1/5) * Zeta(5)^(1/10) / (2^(91/120) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (1749600000000*Zeta(5)^3) + 49 * Pi^8 * Zeta(3) / (2700000 * Zeta(5)^2) - Zeta(3)^2 / (25*Zeta(5)) + (-343 * Pi^12 / (405000000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (750 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (180000 * 2^(3/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + 3^(1/5) * Zeta(3) / (2^(3/5) * (5*Zeta(5))^(2/5))) * n^(2/5) - 7*Pi^4 / (180 * 2^(2/5) * 3^(1/5) * (5*Zeta(5))^(3/5)) * n^(3/5) + 5*3^(2/5) * ((5*Zeta(5))/2)^(1/5)/4 * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.
%t A258345 nmax=40; CoefficientList[Series[Product[(1+x^k)^(k*(k-1)*(k-2)),{k,1,nmax}],{x,0,nmax}],x]
%Y A258345 Cf. A248882, A028377, A258341, A258342, A258343, A258344, A258346, A258351.
%K A258345 nonn
%O A258345 0,4
%A A258345 _Vaclav Kotesovec_, May 27 2015