cp's OEIS Frontend

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

%I A320563 #9 Oct 15 2018 22:19:54
%S A320563 1,1,4,13,41,125,374,1103,3213,9259,26430,74806,210095,585890,1623240,
%T A320563 4470232,12241799,33349751,90410255,243977941,655553258,1754265279,
%U A320563 4676358086,12420299846,32873598566,86721264126,228051843891,597905347237,1563071037798,4074973824099
%N A320563 Expansion of Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^k.
%C A320563 First differences of the binomial transform of A000219.
%H A320563 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A320563 G.f.: exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)).
%F A320563 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
%p A320563 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
%t A320563 nmax = 29; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A320563 nmax = 29; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]
%Y A320563 Cf. A000219, A001157, A103446, A218482, A294500, A320564.
%K A320563 nonn
%O A320563 0,3
%A A320563 _Ilya Gutkovskiy_, Oct 15 2018