A239162 Number of partitions of 3^n into parts that are at most n with at least one part of each size.

0, 1, 4, 48, 3042, 1067474, 2215932130, 29012104252380, 2526293243761311036, 1525710603023191548743988, 6600334932211428773703751221040, 209705652574790086852527310591449309624, 49907101066058865036206450041083799915221352487

Offset: 0

Author

Alois P. Heinz, Mar 11 2014

Keywords

• nonn

Examples

			a(2) = 4: 22221, 222111, 2211111, 21111111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..49
• 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=3 of A238012.

Programs

Mathematica

maxExponent = 30; 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[3^n - n(n+1)/2 + 1] // Round]; Table[an = a[n];
Print[n, " ", an]; an, {n, 0, 12}] (* Jean-François Alcover, Nov 15 2018 *)

Formula

a(n) = [x^(3^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 3^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015