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Expansion of Product_{k>=1} 1 / (1 - x^k)^(k*(3*k + 2)).
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%I #12 Nov 08 2017 18:06:25

%S 1,5,31,148,667,2754,10823,40393,145085,502780,1690603,5530649,

%T 17658430,55141520,168751779,506933980,1496999360,4350994324,

%U 12460305177,35192973824,98116587875,270220568883,735668636567,1981082952258,5279879097853,13933764841202

%N Expansion of Product_{k>=1} 1 / (1 - x^k)^(k*(3*k + 2)).

%H Robert Israel, <a href="/A294692/b294692.txt">Table of n, a(n) for n = 0..5000</a>

%F a(n) ~ exp(4*Pi*n^(3/4) / (3*5^(1/4)) + 2*Zeta(3) * sqrt(5*n) / Pi^2 - 2*5^(5/4) * Zeta(3)^2 * n^(1/4) / Pi^5 + 200*Zeta(3)^3 / (3*Pi^8) - 3*Zeta(3) / (4*Pi^2) + 1/6) * Pi^(1/6) / (A^2 * 2^(3/2) * 5^(1/6) * n^(2/3)), where A is the Glaisher-Kinkelin constant A074962.

%p N:= 50:

%p S:= series(mul(1/(1-x^k)^(k*(3*k+2)), k=1..N),x,N+1):

%p seq(coeff(S,x,n),n=0..N); # _Robert Israel_, Nov 07 2017

%t nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^(k*(3*k+2)), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A074962, A274998, A278768, A294655, A294667.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Nov 07 2017