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

0,2

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

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

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 16 * exp(-13*Pi/24) * 2^(7/8) * Gamma(3/4)^13 / Pi^(13/4) = A388360. - Simon Plouffe, Sep 15 2025

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 *)

Prog

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

Crossrefs

Cf. 13th column of A144064. - Alois P. Heinz, Oct 17 2008

Sequence in context: A283121 A278555 A282921 * A022641 A000590 A052065

Adjacent sequences: A023008 A023009 A023010 * A023012 A023013 A023014

Keyword

nonn

Author

David W. Wilson

Status

approved