A238001 Number of partitions of n^n into parts that are at most n with at least one part of each size.

0, 1, 1, 48, 109809, 32796849930, 2555847904495965819, 85962759806610904434664386174, 1841132100297745277187328924904656111127054, 34687813181057391872792859998288408847592250236051615502024

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..27

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.

Formula

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

a(n) ~ n^(n*(n-1)) / (n!*(n-1)!) ~ exp(2*n) * n^(n*(n-3)) / (2*Pi). - Vaclav Kotesovec, Jun 05 2015

Example

a(1) = 1: 1.
a(2) = 1: 211.
a(3) = 48: 3333333321, ..., 321111111111111111111111.

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[n^n-n(n+1)/2 + 1] // Round];
Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 9}] (* Jean-François Alcover, Nov 15 2018 *)

Crossrefs

Main diagonal of A238012.

Cf. A236810, A237512, A237998, A237999, A238000.

Sequence in context: A006070 A081262 A340186 * A228143 A008704 A037947

Adjacent sequences: A237998 A237999 A238000 * A238002 A238003 A238004

Keyword

nonn

Author

Alois P. Heinz, Feb 16 2014

Status

approved