A367022 Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = 4^n * hypergeom([1/2, -n - 1, -n], [2, 2], x).

1, 4, 1, 16, 12, 2, 64, 96, 48, 5, 256, 640, 640, 200, 14, 1024, 3840, 6400, 4000, 840, 42, 4096, 21504, 53760, 56000, 23520, 3528, 132, 16384, 114688, 401408, 627200, 439040, 131712, 14784, 429, 65536, 589824, 2752512, 6021120, 6322176, 3161088, 709632, 61776, 1430

Offset

0,2

Links

Table of n, a(n) for n=0..44.

Formula

From Detlef Meya, Nov 20 2023: (Start)

T(n, k) = 4^(n - k)*binomial(n, k)*binomial(n+1, k)*binomial(2*k, k)/(k + 1)^2.

T(n, k) = A001263(n+1, k+1)*4^(n - k)*binomial(2*k, k)/(k + 1). (End)

Example

Triangle T(n, k) starts:
  [0]     1;
  [1]     4,      1;
  [2]    16,     12,       2;
  [3]    64,     96,      48,       5;
  [4]   256,    640,     640,     200,      14;
  [5]  1024,   3840,    6400,    4000,     840,      42;
  [6]  4096,  21504,   53760,   56000,   23520,    3528,    132;
  [7] 16384, 114688,  401408,  627200,  439040,  131712,  14784,   429;
  [8] 65536, 589824, 2752512, 6021120, 6322176, 3161088, 709632, 61776, 1430;

Maple

p := n -> 4^n*hypergeom([1/2, -n - 1, -n], [2, 2], x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..8);

Mathematica

T[n_, k_]:=4^(n-k)*Binomial[n, k]*Binomial[n+1, k]*Binomial[2*k, k]/(k+1)^2; Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (* Detlef Meya, Nov 20 2023 *)

Crossrefs

Cf. A038845 (column 1), A128088, A005802, A246513, A001263.

Sequence in context: A292922 A117438 A075499 * A099394 A269698 A059991

Adjacent sequences: A367019 A367020 A367021 * A367023 A367024 A367025

Keyword

nonn,tabl

Author

Peter Luschny, Nov 06 2023

Status

approved