OFFSET
0,8
COMMENTS
Also matrices with k rows of nonnegative integers with distinct positive column sums and total element sum n.
A(2,2) = 3: (matrices and corresponding marked compositions are given)
[1] [2] [0]
[1] [0] [2]
2ab, 2aa, 2bb.
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
FORMULA
A(n,k) = Sum_{i=0..k} C(k,i) * A261836(n,k-i).
EXAMPLE
A(3,2) = 16: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a2aa, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1b2bb.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, ...
0, 1, 3, 6, 10, 15, 21, 28, ...
0, 3, 16, 46, 100, 185, 308, 476, ...
0, 3, 21, 75, 195, 420, 798, 1386, ...
0, 5, 50, 231, 736, 1876, 4116, 8106, ...
0, 11, 205, 1414, 6032, 19320, 51114, 117936, ...
0, 13, 292, 2376, 11712, 42610, 126288, 322764, ...
MAPLE
b:= proc(n, i, p, k) option remember;
`if`(i*(i+1)/2<n, 0, `if`(n=0, p!, b(n, i-1, p, k)+
`if`(i>n, 0, b(n-i, i-1, p+1, k)*binomial(i+k-1, k-1))))
end:
A:= (n, k)-> b(n$2, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
b[n_, i_, p_, k_] := b[n, i, p, k] = If[i*(i+1)/2 < n, 0, If[n == 0, p!, b[n, i-1, p, k] + If[i>n, 0, b[n-i, i-1, p+1, k]*Binomial[i+k-1, k-1]]]]; A[n_, k_] := b[n, n, 0, k]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 16 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 02 2015
STATUS
approved