A239165 Number of partitions of 6^n into parts that are at most n with at least one part of each size.

0, 1, 17, 3781, 14942231, 1264608203048, 2555847904495965819, 132574244496779071303074376, 185560862224740635595130202984468935, 7271076505438083132065943012753686950455454055, 8205115354631567886718289443554629632451344416164686337673

Offset: 0

Author

Alois P. Heinz, Mar 11 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..37
• 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])

Crossrefs

Column k=6 of A238012.

Programs

Mathematica

maxExponent = 50; 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[6^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^(6^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 6^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015