A258347 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k+1)).

1, 2, 9, 28, 88, 250, 708, 1894, 4988, 12718, 31839, 77952, 187771, 444526, 1037522, 2387670, 5426996, 12188774, 27079379, 59541078, 129663636, 279801102, 598620511, 1270300142, 2674874760, 5591124784, 11605082733, 23926811840, 49016020317, 99798382290

Offset

0,2

Links

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

Formula

a(n) ~ Pi^(1/12) / (2^(3/2) * 15^(7/48) * n^(31/48)) * exp(Zeta'(-1) - Zeta(3) / (4*Pi^2) + 75*Zeta(3)^3 / Pi^8 - 15^(5/4) * Zeta(3)^2 / (2*Pi^5) * n^(1/4) + sqrt(15) * Zeta(3) / Pi^2 * sqrt(n) + 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_2(k) + sigma_3(k))*x^k/k). - Ilya Gutkovskiy, Aug 22 2018

Mathematica

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

Crossrefs

Cf. A000294, A023871, A258341, A258348, A258349, A258350, A258351, A258352.

Sequence in context: A192693 A368217 A360479 * A323957 A026087 A109188

Adjacent sequences: A258344 A258345 A258346 * A258348 A258349 A258350

Keyword

nonn

Author

Vaclav Kotesovec, May 27 2015

Status

approved