A309267 Expansion of (1 + x) * Product_{k>=1} 1/(1 - x^k)^k.

1, 2, 4, 9, 19, 37, 72, 134, 246, 442, 782, 1359, 2338, 3964, 6652, 11046, 18176, 29631, 47935, 76931, 122608, 194072, 305269, 477258, 741977, 1147227, 1764778, 2701403, 4115892, 6242846, 9428575, 14181272, 21245738, 31708402, 47150928, 69867001, 103176007, 151864745, 222821779

Offset: 0

Author

Ilya Gutkovskiy, Jul 20 2019

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000219, A052816, A084376, A091360, A191659, A277963.

Programs

Maple

G:= (1+x)/mul((1-x^k)^k,k=1..100):
S:= series(G,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Dec 01 2020

Mathematica

nmax = 38; CoefficientList[Series[(1 + x) Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[2, k] a[n - k], {k, 1, n}]/n; Table[a[n] + a[n - 1], {n, 0, 38}]

Formula

a(n) = A000219(n) + A000219(n-1).

a(n) ~ Zeta(3)^(7/36) * 2^(25/36) * exp(3 * Zeta(3)^(1/3) * (n/2)^(2/3) + 1/12) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jul 20 2019