A023011 Number of partitions of n into parts of 13 kinds.

1, 13, 104, 637, 3276, 14820, 60697, 229372, 810654, 2706366, 8600501, 26173966, 76654656, 216903064, 594973106, 1586553501, 4122693185, 10461067253, 25967050382, 63154957281, 150708128116, 353304272945, 814564136529, 1848834255034, 4134822087942

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

a(n) is Euler transform of A010852. - Alois P. Heinz, Oct 17 2008

Links

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

Crossrefs

Cf. 13th 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*13, 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]^13, {x, 0, 30}], x] (* Indranil Ghosh, Mar 27 2017 *)

PARI

Vec(1/eta(x)^13 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

Formula

a(0) = 1, a(n) = (13/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 = 13. - Vaclav Kotesovec, Jun 28 2025