A337464 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt(2 / ( (1+2*(k-4)*x+((k+4)*x)^2) * (1-(k+4)*x+sqrt(1+2*(k-4)*x+((k+4)*x)^2)) )).

1, 1, 6, 1, 5, 30, 1, 4, 11, 140, 1, 3, -6, -29, 630, 1, 2, -21, -120, -365, 2772, 1, 1, -34, -139, -266, -1409, 12012, 1, 0, -45, -92, 531, 2520, -155, 51480, 1, -1, -54, 15, 1654, 6489, 17380, 29485, 218790, 1, -2, -61, 176, 2755, 4828, -9723, -13104, 170035, 923780

Offset

0,3

Links

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

Formula

T(n,k) = Sum_{j=0..n} (-k)^(n-j) * binomial(2*j,j) * binomial(2*n+1,2*j).

T(0,k) = 1, T(1,k) = 6-k and n * (2*n+1) * (4*n-3) * T(n,k) = (4*n-1) * (-4*(k-4)*n^2+2*(k-4)*n+k-2) * T(n-1,k) - (k+4)^2 * (n-1) * (2*n-1) * (4*n+1) * T(n-2,k) for n > 1. - Seiichi Manyama, Aug 29 2020

Example

Square array begins:
     1,     1,    1,    1,    1,     1, ...
     6,     5,    4,    3,    2,     1, ...
    30,    11,   -6,  -21,  -34,   -45, ...
   140,   -29, -120, -139,  -92,    15, ...
   630,  -365, -266,  531, 1654,  2755, ...
  2772, -1409, 2520, 6489, 4828, -5853, ...

Mathematica

T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-j)] * Binomial[2*j, j] * Binomial[2*n+1, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2021 *)

Prog

(PARI) {T(n, k) = sum(j=0, n, (-k)^(n-j)*binomial(2*j, j)*binomial(2*n+1, 2*j))}

Crossrefs

Columns k=0..4 give A002457, A337394, A337466, A337467, A337397.

Main diagonal gives A337465.

Cf. A337369, A337419.

Sequence in context: A011439 A094774 A231925 * A193239 A023406 A138116

Adjacent sequences: A337461 A337462 A337463 * A337465 A337466 A337467

Keyword

sign,tabl

Author

Seiichi Manyama, Aug 28 2020

Status

approved