A258349 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)/2).

1, 0, 1, 3, 7, 13, 28, 52, 107, 203, 396, 741, 1409, 2596, 4813, 8777, 15972, 28737, 51553, 91644, 162288, 285377, 499653, 869758, 1507615, 2599974, 4465606, 7635607, 13005252, 22061424, 37287395, 62788012, 105365891, 176211393, 293741195, 488101711, 808604106

Offset: 0

Author

Vaclav Kotesovec, May 27 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000294, A023871, A027999, A258347, A258348, A258350, A258351, A258352.

Programs

Mathematica

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)/2),{k,1,nmax}],{x,0,nmax}],x]

SageMath

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n,2))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

Formula

a(n) ~ 1 / (2^(155/96) * 15^(11/96) * Pi^(1/24) * n^(59/96)) * exp(-Zeta'(-1)/2 - Zeta(3) / (8*Pi^2) - 75*Zeta(3)^3 / (2*Pi^8) - 15^(5/4) * Zeta(3)^2 / (2^(7/4) * Pi^5) * n^(1/4) - sqrt(15/2) * Zeta(3) / Pi^2 * sqrt(n) + 2^(7/4)*Pi / (3*15^(1/4)) * n^(3/4)), where Zeta(3) = A002117, Zeta'(-1) = A084448 = 1/12 - log(A074962).

G.f.: exp(Sum_{k>=1} (sigma_3(k) - sigma_2(k))*x^k/(2*k)). - Ilya Gutkovskiy, Aug 22 2018