A023014 Number of partitions of n into parts of 16 kinds.

1, 16, 152, 1088, 6460, 33440, 155584, 663936, 2636326, 9845040, 34861152, 117809728, 381946360, 1193074144, 3603543040, 10556065152, 30068145905, 83466484112, 226236086512, 599785472000, 1557643542308, 3967888347232, 9926348625408, 24413219138816

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

a(n) is Euler transform of A010855. - 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. 16th 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*16, 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]^16, {x, 0, 30}], x] (* Indranil Ghosh, Mar 27 2017 *)

PARI

Vec(1/eta(x)^16 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

Formula

G.f.: Product_{m>=1} 1/(1-x^m)^16.

a(0) = 1, a(n) = (16/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 = 16. - Vaclav Kotesovec, Jun 28 2025