A023019 Number of partitions of n into parts of 21 kinds.

1, 21, 252, 2233, 16170, 100926, 560945, 2837418, 13266099, 57994475, 239170239, 937026279, 3507380170, 12601619226, 43628951025, 146036139347, 473924014599, 1494785958435, 4591920193357, 13764656869425, 40328218603134

Offset: 0

Author

David W. Wilson, Jun 14 1998

Keywords

• nonn

Comments

a(n) is the Euler transform of A010860. - 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

Cf. 21st 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*21, 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]^21, {x, 0, 30}], x] (* Indranil Ghosh, Mar 27 2017 *)

PARI

Vec(1/eta(x)^21 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

Formula

a(0) = 1, a(n) = (21/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 = 21. - Vaclav Kotesovec, Jun 28 2025