A238012 Number A(n,k) of partitions of k^n into parts that are at most n with at least one part of each size; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 4, 2, 0, 0, 0, 1, 7, 48, 9, 0, 0, 0, 1, 12, 310, 3042, 119, 0, 0, 0, 1, 17, 1240, 109809, 1067474, 4935, 0, 0, 0, 1, 24, 3781, 1655004, 370702459, 2215932130, 596763, 0, 0, 0, 1, 31, 9633, 14942231, 32796849930, 13173778523786, 29012104252380, 211517867, 0, 0

Offset: 0

Author

Alois P. Heinz, Feb 16 2014

Keywords

• nonn
• tabl

Comments

In general, column k>=2 is asymptotic to k^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

Examples

			Square array A(n,k) begins:
  0, 0,   0,       0,         0,           0, ...
  0, 1,   1,       1,         1,           1, ...
  0, 0,   1,       4,         7,          12, ...
  0, 0,   2,      48,       310,        1240, ...
  0, 0,   9,    3042,    109809,     1655004, ...
  0, 0, 119, 1067474, 370702459, 32796849930, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..43, flattened
• 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

Columns k=0-10 give: A000004, A063524, A237999, A239162, A239163, A239164, A239165, A239166, A239167, A239168, A239169.

Rows n=0-2 give: A000004, A057427, A074148(k-1) for k>1.

Main diagonal gives A238001.

Cf. A238010.

Programs

Mathematica

A[0, 0] = 0;
A[n_, k_] := SeriesCoefficient[Product[1/(1-x^j), {j, 1, n}], {x, 0, k^n - n(n+1)/2}];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Aug 18 2018, after Alois P. Heinz *)

Formula

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