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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.
12
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
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
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
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Feb 16 2014
STATUS
approved