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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.
15
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
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
Rows n=0-2 give: A000004, A000012, A080827.
Main diagonal gives A238000.
Sequence in context: A336111 A244657 A072024 * A370772 A011354 A143119
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Feb 16 2014
STATUS
approved