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

0,2

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

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

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 64 * exp(-2*Pi/3) * Gamma(3/4)^16 / Pi^4 = A388363. - Simon Plouffe, Sep 15 2025

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 *)

Prog

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

Crossrefs

Cf. 16th column of A144064. - Alois P. Heinz, Oct 17 2008

Sequence in context: A240424 A341391 A225915 * A073384 A022644 A297090

Adjacent sequences: A023011 A023012 A023013 * A023015 A023016 A023017

Keyword

nonn

Author

David W. Wilson

Status

approved