A362847 Triangle read by rows, T(n, k) = 4^k * Gamma(n + k + 1/2) / Gamma(n - k + 1/2).

1, 1, 3, 1, 15, 105, 1, 35, 945, 10395, 1, 63, 3465, 135135, 2027025, 1, 99, 9009, 675675, 34459425, 654729075, 1, 143, 19305, 2297295, 218243025, 13749310575, 316234143225, 1, 195, 36465, 6235515, 916620705, 105411381075, 7905853580625, 213458046676875

Offset

0,3

Links

Table of n, a(n) for n=0..35.

Formula

T(n ,k ) = (2*(n + k) - 1)!!/(2*(n - k) - 1)!!; 0 <= n <= k. - Detlef Meya, Oct 09 2023

Example

[0] 1;
[1] 1,   3;
[2] 1,  15,   105;
[3] 1,  35,   945,   10395;
[4] 1,  63,  3465,  135135,   2027025;
[5] 1,  99,  9009,  675675,  34459425,   654729075;
[6] 1, 143, 19305, 2297295, 218243025, 13749310575, 316234143225;

Maple

T := (n, k) -> 4^k * GAMMA(n + k + 1/2) / GAMMA(n - k + 1/2):
seq(seq(T(n, k), k = 0..n), n = 0..7);

Mathematica

T[n_, k_]:=(2*(n+k)-1)!!/(2*(n-k)-1)!!; Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]] (* Detlef Meya, Oct 09 2023 *)

Crossrefs

Cf. A362848 (row sums), A000466 (column 1), A101485 (main diagonal).

Sequence in context: A160628 A160604 A160616 * A168319 A326374 A143565

Adjacent sequences: A362844 A362845 A362846 * A362848 A362849 A362850

Keyword

nonn,tabl

Author

Peter Luschny, May 05 2023

Status

approved