A336727 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} (-k)^(n-j) * binomial(n,j) * binomial(n,j-1) for n > 0.

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 0, 1, 1, 1, -3, 1, 5, 2, 1, 1, 1, -4, 5, 10, -3, 0, 1, 1, 1, -5, 11, 9, -38, -21, -5, 1, 1, 1, -6, 19, -4, -103, 28, 51, 0, 1, 1, 1, -7, 29, -35, -174, 357, 289, 41, 14, 1, 1, 1, -8, 41, -90, -203, 1176, -131, -1262, -391, 0, 1

Offset

0,18

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) / (1 + k * x * A_k(x)).

A_k(x) = 2/(1 - (k+1)*x + sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2)).

T(n, k) = Sum_{j=0..n} (-k)^j * (k+1)^(n-j) * binomial(n,j) * binomial(n+j,n)/(j+1).

(n+1) * T(n,k) = -(k-1) * (2*n-1) * T(n-1,k) - (k+1)^2 * (n-2) * T(n-2,k) for n>1. - Seiichi Manyama, Aug 08 2020

Example

  1,  1,   1,   1,    1,    1,    1, ...
  1,  1,   1,   1,    1,    1,    1, ...
  1,  0,  -1,  -2,   -3,   -4,   -5, ...
  1, -1,  -1,   1,    5,   11,   19, ...
  1,  0,   5,  10,    9,   -4,  -35, ...
  1,  2,  -3, -38, -103, -174, -203, ...
  1,  0, -21,  28,  357, 1176, 2575, ...

Mathematica

T[0, k_] := 1; T[n_, k_] := Sum[If[k == 0, Boole[n == j], (-k)^(n - j)] * Binomial[n, j] * Binomial[n , j - 1], {j, 1, n}] / n; Table[T[k, n- k], {n, 0, 11}, {k, 0, n}] //Flatten (* Amiram Eldar, Aug 02 2020 *)

Prog

(PARI) {T(n, k) = if(n==0, 1, sum(j=1, n, (-k)^(n-j)*binomial(n, j)*binomial(n, j-1))/n)}
(PARI) {T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A/(1+k*x*A)); polcoef(A, n)}
(PARI) {T(n, k) = sum(j=0, n, (-k)^j*(k+1)^(n-j)*binomial(n, j)*binomial(n+j, n)/(j+1))}

Crossrefs

Columns k=0-3 give: A000012, A090192, (-1)^n * A154825(n), A336729.

Main diagonal gives A336728.

Cf. A243631, A307884, A336708, A336709.

Sequence in context: A273693 A219967 A060505 * A316101 A211452 A035188

Adjacent sequences: A336724 A336725 A336726 * A336728 A336729 A336730

Keyword

sign,tabl

Author

Seiichi Manyama, Aug 02 2020

Status

approved