OFFSET
0,6
FORMULA
T(0,k) = 1 and T(n,k) = binomial(n,k) * Sum_{j=k+1..n} binomial(n-k,j-k) * T(n-j,k) for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * Stirling2(n-k*j,j)/(k!^j * (n-k*j)!).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, 0, ...
3, 2, 0, 0, 0, 0, 0, ...
13, 3, 3, 0, 0, 0, 0, ...
75, 28, 6, 4, 0, 0, 0, ...
541, 125, 10, 10, 5, 0, 0, ...
4683, 1146, 195, 20, 15, 6, 0, ...
PROG
(PARI) T(n, k) = n!*sum(j=0, n\(k+1), j!*stirling(n-k*j, j, 2)/(k!^j*(n-k*j)!));
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jul 13 2022
STATUS
approved