OFFSET
0,12
COMMENTS
Column k is the diagonal of the rational function 1 / (1 - Sum_{j=1..k} x_j + Product_{j=1..k} x_j) for k>0.
FORMULA
G.f. of column k: Sum_{j>=0} (k*j)!/j!^k * x^j / (1+x)^(k*j+1).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 0, 1, 5, 23, 119, ...
1, 0, 1, 67, 2401, 112681, ...
0, 0, 1, 1109, 347279, 166923119, ...
1, 0, 1, 20251, 58370761, 302857024681, ...
0, 0, 1, 391355, 10693893503, 616967236620839, ...
MATHEMATICA
T[n_, k_] := Sum[(-1)^(n - j)*(n + (k - 1)*j)!/(n - j)!/(j!)^k, {j, 0, n} ]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 10 2020 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jul 10 2020
STATUS
approved