OFFSET
0,6
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 - 2 * x * A_k(x)).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
2, 3, 4, 5, 6, 7, 8, ...
6, 11, 19, 30, 44, 61, 81, ...
20, 45, 100, 201, 364, 605, 940, ...
72, 197, 562, 1445, 3249, 6502, 11857, ...
272, 903, 3304, 10900, 30526, 73723, 158034, ...
MATHEMATICA
T[0, k_] := 1; T[n_, k_] := Sum[2^(n - j) * 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, 2^(n-j)*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/(1-2*x*A)); polcoef(A, n)}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Aug 01 2020
STATUS
approved