A023008 Number of partitions of n into parts of 9 kinds.

1, 9, 54, 255, 1035, 3753, 12483, 38709, 113265, 315445, 841842, 2164185, 5382276, 12994290, 30543210, 70066809, 157199805, 345552183, 745377215, 1579915080, 3294664578, 6766656315, 13700560491, 27370137195, 53991639855, 105242612526, 202837976145

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

a(n) is Euler transform of A010734. - Alois P. Heinz, Oct 17 2008

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
• P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, A. M. Läuchli, Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
• N. J. A. Sloane, Transforms
• Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Crossrefs

Cf. 9th 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*9, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

Mathematica

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^9,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

Formula

a(n) ~ 3^(5/2) * exp(Pi * sqrt(6*n)) / (256 * n^3). - Vaclav Kotesovec, Feb 28 2015

a(0) = 1, a(n) = (9/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017

G.f.: exp(9*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018