OFFSET
0,6
FORMULA
E.g.f. of column k: 1/(1 - x*exp(k*x)).
T(0,k) = 1 and T(n,k) = n * Sum_{j=0..n-1} k^(n-1-j) * binomial(n-1,j) * T(j,k) for n > 0.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
2, 4, 6, 8, 10, 12, ...
6, 21, 42, 69, 102, 141, ...
24, 148, 392, 780, 1336, 2084, ...
120, 1305, 4600, 11145, 22200, 39145, ...
MATHEMATICA
T[n_, k_] := n!*(1 + Sum[(k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 19 2022 *)
PROG
(PARI) T(n, k) = n!*sum(j=0, n, (k*(n-j))^j/j!);
(PARI) T(n, k) = if(n==0, 1, n*sum(j=0, n-1, k^(n-1-j)*binomial(n-1, j)*T(j, k)));
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Feb 19 2022
STATUS
approved