A258352 - OEIS

%I #7 Nov 11 2020 09:01:14

%S 1,0,0,1,4,10,21,39,76,145,294,581,1169,2276,4435,8494,16237,30768,

%T 58221,109466,205223,382658,710808,1314091,2420437,4439753,8115645,

%U 14781062,26833241,48550863,87575527,157480827,282362462,504819198,900058558,1600424247

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

%H Vaclav Kotesovec, <a href="/A258352/b258352.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ Zeta(5)^(379/3600) / (2^(521/1800) * sqrt(5*Pi) * n^(2179/3600)) * exp(Zeta'(-1)/3 + Zeta(3)/(8*Pi^2) - Pi^16 / (3110400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (216000 * Zeta(5)^2) - Zeta(3)^2/(90*Zeta(5)) + Zeta'(-3)/6 + (-Pi^12 / (10800000 * 2^(2/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (900 * 2^(2/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (36000 * 2^(4/5) * Zeta(5)^(7/5)) + Zeta(3) / (3 * 2^(4/5) * Zeta(5)^(2/5))) * n^(2/5) - Pi^4 / (180 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5 * Zeta(5)^(1/5) / 2^(8/5) * 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)/6),{k,1,nmax}],{x,0,nmax}],x]

%o (SageMath) # uses[EulerTransform from A166861]

%o b = EulerTransform(lambda n: binomial(n, 3))

%o print([b(n) for n in range(37)]) # _Peter Luschny_, Nov 11 2020

%Y Cf. A000335, A023872, A258346, A258347, A258348, A258349, A258350, A258351.

%K nonn

%O 0,5

%A _Vaclav Kotesovec_, May 27 2015