A023009 Number of partitions of n into parts of 10 kinds.

1, 10, 65, 330, 1430, 5512, 19415, 63570, 195910, 573430, 1605340, 4322110, 11240645, 28341730, 69488650, 166096270, 387890625, 886698670, 1987322415, 4373271870, 9461022285, 20144164040, 42254620785, 87398226990, 178396331100, 359618772656, 716409453320

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

a(n) is Euler transform of A010692. - 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)
• Index entries for expansions of Product_{k >= 1} (1-x^k)^m
• N. J. A. Sloane, Transforms

Crossrefs

Cf. 10th 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*10, 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)^10,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

Formula

G.f.: Product_{m>=1} 1/(1-x^m)^10.

a(n) ~ 5^(11/4) * exp(2 * Pi * sqrt(5*n/3)) / (64 * 3^(11/4) * n^(13/4)). - Vaclav Kotesovec, Feb 28 2015

a(0) = 1, a(n) = (10/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017

G.f.: exp(10*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018