A023021 Number of partitions of n into parts of 23 kinds.

1, 23, 299, 2852, 22126, 147407, 871838, 4680845, 23177583, 107100903, 466066181, 1923780950, 7576060505, 28601630657, 103928814438, 364712523658, 1239637963457, 4091266414235, 13139808783725, 41145568478988, 125833948024603, 376417734772625, 1102878148698235

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

a(n) is Euler transform of A010862. - Alois P. Heinz, Oct 17 2008

Convolved with A000041 = A006922. - Gary W. Adamson, Jun 09 2009

Links

• Alois P. Heinz, 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

Cf. 23rd column of A144064. - Alois P. Heinz, Oct 17 2008

Cf. A006922, A000041. - Gary W. Adamson, Jun 09 2009

Programs

Maple

with (numtheory): a:= proc(n) option remember; `if`(n=0, 1, add (add (d*23, 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[1/QPochhammer[q]^23 + O[q]^30, q] (* Jean-François Alcover, Dec 03 2015 *)

PARI

Vec(1/eta(x)^23 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

Formula

G.f.: Product_{m>=1} 1/(1-x^m)^23.

a(0) = 1, a(n) = (23/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 = 23. - Vaclav Kotesovec, Jun 28 2025