A023018 Number of partitions of n into parts of 20 kinds.

1, 20, 230, 1960, 13685, 82524, 443870, 2175800, 9869990, 41907380, 168012824, 640438680, 2334121995, 8171039800, 27580783270, 90058003200, 285253928790, 878572253720, 2636748302650, 7725084195240, 22130265931900, 62079251390180

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

a(n) is Euler transform of A010859. - Alois P. Heinz, Oct 17 2008

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, Transforms
• Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Crossrefs

20th column of A144064. - Alois P. Heinz, Oct 17 2008

Programs

Maple

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*20, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

Mathematica

CoefficientList[Series[1/QPochhammer[x]^20, {x, 0, 30}], x] (* Indranil Ghosh, Mar 27 2017 *)

PARI

Vec(1/eta(x)^20 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

Formula

G.f.: Product_{m>=1} 1/(1-x^m)^20.

a(0) = 1, a(n) = (20/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017

a(n) ~ m^((m+1)/4) * exp(Pi*sqrt(2*m*n/3)) / (2^((3*m+5)/4) * 3^((m+1)/4) * n^((m+3)/4)) * (1 - ((9+Pi^2)*m^2+36*m+27) / (24*Pi*sqrt(6*m*n))), set m = 20. - Vaclav Kotesovec, Jun 28 2025