A238010 Number A(n,k) of partitions of k^n into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 5, 10, 1, 1, 0, 1, 9, 75, 64, 1, 1, 0, 1, 13, 374, 4410, 831, 1, 1, 0, 1, 19, 1365, 123464, 1366617, 26207, 1, 1, 0, 1, 25, 3997, 1736385, 393073019, 2559274110, 2239706, 1, 1

Offset

0,13

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])

Formula

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

Example

A(3,2) = 10: 332, 2222, 3221, 3311, 22211, 32111, 221111, 311111, 2111111, 11111111.
A(2,3) = 5: 22221, 222111, 2211111, 21111111, 111111111.
A(2,4) = 9: 22222222, 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111, 1111111111111111.
Square array A(n,k) begins:
  0, 0,   0,       0,         0,           0, ...
  1, 1,   1,       1,         1,           1, ...
  1, 1,   3,       5,         9,          13, ...
  1, 1,  10,      75,       374,        1365, ...
  1, 1,  64,    4410,    123464,     1736385, ...
  1, 1, 831, 1366617, 393073019, 33432635477, ...

Mathematica

A[n_, k_] := SeriesCoefficient[Product[1/(1-x^j), {j, 1, n}], {x, 0, k^n}]; A[0, 0] = 0; Table[A[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 17 2017 *)

Crossrefs

Columns k=0+1,2-10 give: A057427, A237998, A238560, A238561, A238562, A238563, A238564, A238565, A238566, A238567.

Rows n=0-2 give: A000004, A000012, A080827.

Main diagonal gives A238000.

Cf. A238012, A238016.

Sequence in context: A336111 A244657 A072024 * A370772 A011354 A143119

Adjacent sequences: A238007 A238008 A238009 * A238011 A238012 A238013

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 16 2014

Status

approved