A023020 Number of partitions of n into parts of 22 kinds.

1, 22, 275, 2530, 18975, 122430, 702328, 3661900, 17627775, 79264900, 335937954, 1351507830, 5191041625, 19125838600, 67862904725, 232671319474, 773027485065, 2494957906100, 7839428942950, 24025993453000, 71941861591215

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

a(n) is Euler transform of A010861. - Alois P. Heinz, Oct 17 2008

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• M. V. Movshev, A formula for the partition function of the beta-gamma system on the cone pure spinors, arXiv preprint arXiv:1602.04673 [hep-th], 2016. [Gives sequence that appears to agree with this one]
• N. J. A. Sloane, Transforms [_Alois P. Heinz_, Oct 17 2008]
• Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Crossrefs

Cf. 22nd 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*22, 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]^22, {x, 0, 30}], x] (* Indranil Ghosh, Mar 27 2017 *)

PARI

Vec(1/eta(x)^22 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

Formula

G.f.: Product_{m>=1} 1/(1-x^m)^22.

a(0) = 1, a(n) = (22/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 = 22. - Vaclav Kotesovec, Jun 28 2025