A239164 Number of partitions of 5^n into parts that are at most n with at least one part of each size.

0, 1, 12, 1240, 1655004, 32796849930, 10743023668660275, 62590747974586286694030, 6826987264035710020018176749475, 14471606032117455546329821353159274382372, 613427607589897771307393494301176209875530879140211

Offset: 0

Author

Alois P. Heinz, Mar 11 2014

Keywords

• nonn

Examples

			a(2) = 12: 2222222222221, 22222222222111, 222222222211111, 2222222221111111, 22222222111111111, 222222211111111111, 2222221111111111111, 22222111111111111111, 222211111111111111111, 2221111111111111111111, 22111111111111111111111, 211111111111111111111111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..40
• A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz), arXiv:1108.4391 [math.CO], 2011.

Crossrefs

Column k=5 of A238012.

Programs

Mathematica

maxExponent = 45; a[0] = 0; a[1] = 1;
a[n_] := Module[{}, aparts = List @@ (Product[1/(1 - x^j), {j, 1, n}] // Apart); cc = aparts + O[x]^maxExponent // CoefficientList[#, x]&; f[k_] = Total[FindSequenceFunction[#, k]& /@ cc];f[5^n - n(n+1)/2 + 1] // Round];
Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 10}] (* Jean-François Alcover, Nov 15 2018 *)

Formula

a(n) = [x^(5^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 5^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015