A023016 Number of partitions of n into parts of 18 kinds.

1, 18, 189, 1482, 9576, 53676, 269325, 1235286, 5256711, 20985272, 79260723, 285139764, 982349361, 3255488082, 10416507579, 32281134120, 97154549289, 284625019800, 813310723925, 2270826800172, 6204926551824, 16615751700618

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

a(n) is Euler transform of A010857. - Alois P. Heinz, Oct 17 2008

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Crossrefs

Cf. 18th 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*18, 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]^18, {x, 0, 30}], x] (* Indranil Ghosh, Mar 27 2017 *)

PARI

Vec(1/eta(x)^18 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

Formula

a(0) = 1, a(n) = (18/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 = 18. - Vaclav Kotesovec, Jun 28 2025