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A258342 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)*(k+2)).

%I A258342 #7 May 28 2015 03:38:22
%S A258342 1,6,39,224,1131,5412,24411,105078,435048,1740312,6755877,25533330,
%T A258342 94205738,340064322,1203313782,4180514846,14279610417,48013553310,
%U A258342 159086287869,519912616614,1677331973910,5345927500226,16843574682291,52494817082952,161923200857711
%N A258342 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)*(k+2)).
%H A258342 Vaclav Kotesovec, <a href="/A258342/b258342.txt">Table of n, a(n) for n = 0..1000</a>
%F A258342 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 A258342 nmax=30; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)*(k+2)),{k,1,nmax}],{x,0,nmax}],x]
%Y A258342 Cf. A248882, A028377, A258341, A258343, A258344, A258345, A258346, A258350.
%K A258342 nonn
%O A258342 0,2
%A A258342 _Vaclav Kotesovec_, May 27 2015