A294692 Expansion of Product_{k>=1} 1 / (1 - x^k)^(k*(3*k + 2)).

1, 5, 31, 148, 667, 2754, 10823, 40393, 145085, 502780, 1690603, 5530649, 17658430, 55141520, 168751779, 506933980, 1496999360, 4350994324, 12460305177, 35192973824, 98116587875, 270220568883, 735668636567, 1981082952258, 5279879097853, 13933764841202

Offset: 0

Author

Vaclav Kotesovec, Nov 07 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..5000

Crossrefs

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

Programs

Maple

N:= 50:
S:= series(mul(1/(1-x^k)^(k*(3*k+2)), k=1..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Nov 07 2017

Mathematica

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

Formula

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.