OFFSET
0,9
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
FORMULA
G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 - x * A_k(x)).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
2, 5, 11, 20, 32, 47, 65, ...
4, 14, 45, 113, 234, 424, 699, ...
9, 42, 197, 688, 1854, 4159, 8192, ...
21, 132, 903, 4404, 15490, 43097, 101538, ...
MATHEMATICA
T[0, k_] := 1; T[n_, k_] := Sum[Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)
PROG
(PARI) {T(n, k) = if(n==0, 1, sum(j=1, n, binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}
(PARI) {T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1-x*A)); polcoef(A, n)}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Aug 01 2020
STATUS
approved