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%I #6 May 28 2015 03:36:48
%S 1,6,45,260,1410,7026,33212,149190,643959,2681020,10820736,42468828,
%T 162566956,608302638,2229485529,8016901068,28324233846,98447346282,
%U 336996263702,1137220855428,3786525025002,12449461237388,40446207528429,129926295916884,412912082761651
%N Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k+1)*(k+2)).
%H Vaclav Kotesovec, <a href="/A258350/b258350.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) ~ (3*Zeta(5))^(79/600) / (2^(21/200) * sqrt(5*Pi) * n^(379/600)) * exp(2*Zeta'(-1) - 3*Zeta(3)/(4*Pi^2) - Pi^16 / (518400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (36000 * Zeta(5)^2) - Zeta(3)^2 / (15*Zeta(5)) + Zeta'(-3) + (Pi^12 / (1800000 * 2^(3/5) * 3^(1/5) * Zeta(5)^(11/5)) - Pi^4 * Zeta(3) / (150 * 2^(3/5) * 3^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (12000 * 2^(1/5) * 3^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(1/5) * (3*Zeta(5))^(2/5))) * n^(2/5) + Pi^4 / (30 * 2^(4/5) * (3*Zeta(5))^(3/5)) * n^(3/5) + 5 * (3*Zeta(5))^(1/5) / 2^(7/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-1) = A084448 = 1/12 - log(A074962), Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4.
%t nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)*(k+2)),{k,1,nmax}],{x,0,nmax}],x]
%Y Cf. A023872, A000335, A258342, A258347, A258348, A258349, A258351, A258352.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, May 27 2015
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