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

0,18

Comments

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

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.

Formula

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

Example

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, ...

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 *)

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.

Sequence in context: A056582 A167891 A105087 * A324802 A320647 A028572

Adjacent sequences: A238009 A238010 A238011 * A238013 A238014 A238015

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 16 2014

Status

approved