A091360 Partial sums of A000219.

1, 2, 5, 11, 24, 48, 96, 182, 342, 624, 1124, 1983, 3462, 5947, 10114, 16993, 28290, 46624, 76225, 123555, 198833, 317627, 504102, 794885, 1246079, 1942112, 3010857, 4643515, 7126749, 10886361, 16555324, 25067633, 37801062, 56776035, 84951990, 126643036, 188127997, 278507781, 410949776, 604437277, 886284200, 1295668181

Offset: 0

Author

Christian G. Bower, Jan 02 2004

Keywords

• nonn

Comments

Convergent of columns of A091355.

Links

• Joerg Arndt and Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (first 500 terms from Joerg Arndt)
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000219, A002117, A074962.

Programs

Mathematica

CoefficientList[Series[1/(1-x)*Product[1/(1-x^k)^k,{k,1,50}],{x,0,50}],x] (* Vaclav Kotesovec, Aug 16 2015 *)

PARI

N=66; x='x+O('x^N); Vec( 1/((1-x)*prod(n=1,N, (1-x^n)^n )) ) \\ Joerg Arndt, Mar 15 2014

Formula

Euler transform of 2, 2, 3, 4, 5, 6, 7, 8, 9, ...

G.f.: 1/( (1-x) * prod(n>=1, (1-x^n)^n ) ). [Joerg Arndt, Mar 15 2014]

From Vaclav Kotesovec, Aug 16 2015: (Start)

a(n) = Sum_{k=0..n} A000219(k).

a(n) ~ (n/(2*Zeta(3)))^(1/3) * A000219(n).

a(n) ~ exp(1/12 + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 2^(23/36) * sqrt(3*Pi) * Zeta(3)^(5/36) * n^(13/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

(End)

G.f.: exp(Sum_{k>=1} (sigma_2(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018