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

0,2

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

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

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 256 * exp(-23*Pi/24) * 2^(5/8) * Gamma(3/4)^23 / Pi^(23/4) = A388370. - Simon Plouffe, Sep 15 2025

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 *)

Prog

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

Crossrefs

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

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

Sequence in context: A125435 A086943 A131273 * A022651 A267266 A077504

Adjacent sequences: A023018 A023019 A023020 * A023022 A023023 A023024

Keyword

nonn

Author

David W. Wilson

Status

approved