A336708 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-1)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

1, 1, 1, 1, 1, -1, 1, 1, 0, 0, 1, 1, 1, -1, 2, 1, 1, 2, 1, 0, -3, 1, 1, 3, 6, 1, 2, -1, 1, 1, 4, 14, 21, 1, 0, 11, 1, 1, 5, 25, 76, 80, 1, -5, -15, 1, 1, 6, 39, 182, 450, 322, 1, 0, -13, 1, 1, 7, 56, 355, 1447, 2818, 1347, 1, 14, 77, 1, 1, 8, 76, 611, 3532, 12175, 18352, 5798, 1, 0, -86

Offset

0,15

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,  0, 1,   2,    3,     4,     5, ...
   0, -1, 1,   6,   14,    25,    39, ...
   2,  0, 1,  21,   76,   182,   355, ...
  -3,  2, 1,  80,  450,  1447,  3532, ...
  -1,  0, 1, 322, 2818, 12175, 37206, ...

Mathematica

T[0, k_] := 1; T[n_, k_] := Sum[(-1)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)

Prog

(PARI) {T(n, k) = if(n==0, 1, sum(j=1, n, (-1)^(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+x*A)); polcoef(A, n)}

Crossrefs

Columns k=0-3 give: A007440, A090192, A000012, A106228.

Main diagonal gives A336713.

Cf. A336575, A336706, A336707, A336709.

Sequence in context: A307016 A375383 A143158 * A308424 A317489 A345647

Adjacent sequences: A336705 A336706 A336707 * A336709 A336710 A336711

Keyword

sign,tabl

Author

Seiichi Manyama, Aug 01 2020

Extensions

Typo in name corrected by Georg Fischer, Sep 19 2023

Status

approved