OFFSET
0,8
COMMENTS
A(n,k) is the number of nonnegative integer matrices with k columns and any number of nonzero rows with column sums n. - Andrew Howroyd, Jan 23 2020
LINKS
Alois P. Heinz, Antidiagonals n = 0..48, flattened
FORMULA
A(n,k) = Sum_{j=0..n*k} binomial(j+n-1,n)^k * Sum_{i=j..n*k} (-1)^(i-j) * binomial(i,j). - Andrew Howroyd, Jan 23 2020
EXAMPLE
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 3, 13, 75, 541, ...
1, 2, 26, 818, 47834, 4488722, ...
1, 4, 252, 64324, 42725052, 58555826884, ...
1, 8, 2568, 5592968, 44418808968, 936239675880968, ...
1, 16, 26928, 515092048, 50363651248560, 16811849850663255376, ...
MAPLE
A:= (n, k)-> `if`(k=0, 1, ceil(2^(n-1))*add(add((-1)^i*
binomial(j, i)*binomial(j-i, n)^k, i=0..j), j=0..k*n)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
A[n_, k_] := Sum[If[k == 0, 1, Binomial[j+n-1, n]^k] Sum[(-1)^(i-j)* Binomial[i, j], {i, j, n k}], {j, 0, n k}];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Nov 04 2021 *)
PROG
(PARI) T(n, k)={my(m=n*k); sum(j=0, m, binomial(j+n-1, n)^k*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))} \\ Andrew Howroyd, Jan 23 2020
CROSSREFS
Main diagonal gives A316677.
KEYWORD
AUTHOR
Alois P. Heinz, Jul 10 2018
STATUS
approved