A320563 Expansion of Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^k.

1, 1, 4, 13, 41, 125, 374, 1103, 3213, 9259, 26430, 74806, 210095, 585890, 1623240, 4470232, 12241799, 33349751, 90410255, 243977941, 655553258, 1754265279, 4676358086, 12420299846, 32873598566, 86721264126, 228051843891, 597905347237, 1563071037798, 4074973824099

Offset: 0

Author

Ilya Gutkovskiy, Oct 15 2018

Keywords

• nonn

Comments

First differences of the binomial transform of A000219.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000219, A001157, A103446, A218482, A294500, A320564.

Programs

Maple

seq(coeff(series(mul((1-x^k/(1-x)^k)^(-k),k=1..n),x,n+1), x, n), n = 0 .. 29); # Muniru A Asiru, Oct 15 2018

Mathematica

nmax = 29; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)).

a(n) ~ Zeta(3)^(7/36) * 2^(n - 11/18) * exp(3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + Zeta(3)^(2/3) * n^(1/3) / 2^(5/3) + (1 - Zeta(3))/12) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 15 2018