A331791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2*k*x + ((k-2)*x)^2 + (1 - k*x) * sqrt(1 - 2*k*x + ((k-2)*x)^2)).

1, 1, 0, 1, 2, -3, 1, 4, 3, 0, 1, 6, 15, 4, 10, 1, 8, 33, 56, 5, 0, 1, 10, 57, 180, 210, 6, -35, 1, 12, 87, 400, 985, 792, 7, 0, 1, 14, 123, 740, 2810, 5418, 3003, 8, 126, 1, 16, 165, 1224, 6285, 19824, 29953, 11440, 9, 0, 1, 18, 213, 1876, 12130, 53550, 140497, 166344, 43758, 10, -462

Offset

0,5

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Formula

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

n * (n+2) * T(n,k) = (n+1) * (k * (2*n+1) * T(n-1,k) - (k-2)^2 * n * T(n-2,k)) for n > 1.

T(n,k) = Sum_{j=0..floor(n/2)} (k-1)^j * k^(n-2*j) * binomial(n+1,n-2*j) * binomial(2*j+1,j). - Seiichi Manyama, Aug 24 2025

From Seiichi Manyama, Aug 27 2025: (Start)

T(n,k) = [x^n] (1+k*x+(k-1)*x^2)^(n+1).

For k != 1, e.g.f. of column k: exp(k*x) * BesselI(1, 2*sqrt(k-1)*x) / sqrt(k-1), with offset 1. (End)

Example

Square array begins:
   1, 1,   1,    1,     1,     1, ...
   0, 2,   4,    6,     8,    10, ...
  -3, 3,  15,   33,    57,    87, ...
   0, 4,  56,  180,   400,   740, ...
  10, 5, 210,  985,  2810,  6285, ...
   0, 6, 792, 5418, 19824, 53550, ...

Mathematica

T[n_, k_] := Sum[If[k==1 && j==0, 1, (k-1)^j] * Binomial[n + 1, j] * Binomial[n + 1, j + 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 05 2021 *)

Crossrefs

Columns k=1..5 give A000027(n+1), A001791(n+1), A050151(n+1), A331792, A331793.

T(n,n+1) gives A331794.

Cf. A307883, A331511, A331514, A331795.

Sequence in context: A340313 A120873 A125161 * A365327 A125933 A255054

Adjacent sequences: A331788 A331789 A331790 * A331792 A331793 A331794

Keyword

sign,tabl

Author

Seiichi Manyama, Jan 26 2020

Status

approved