A367178 Triangle read by rows. T(n, k) = binomial(n, k)^2 * CatalanNumber(k).

1, 1, 1, 1, 4, 2, 1, 9, 18, 5, 1, 16, 72, 80, 14, 1, 25, 200, 500, 350, 42, 1, 36, 450, 2000, 3150, 1512, 132, 1, 49, 882, 6125, 17150, 18522, 6468, 429, 1, 64, 1568, 15680, 68600, 131712, 103488, 27456, 1430, 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862

Offset: 0

Author

Peter Luschny, Nov 07 2023

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1,  1;
  [2] 1,  4,    2;
  [3] 1,  9,   18,     5;
  [4] 1, 16,   72,    80,     14;
  [5] 1, 25,  200,   500,    350,     42;
  [6] 1, 36,  450,  2000,   3150,   1512,    132;
  [7] 1, 49,  882,  6125,  17150,  18522,   6468,    429;
  [8] 1, 64, 1568, 15680,  68600, 131712, 103488,  27456,   1430;
  [9] 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862;
		

Crossrefs

Cf. A086618 (row sums), A186415 (central column), A000108 (main diagonal).

Cf. A098474, A367022, A367023, A387024, A387024, A367177.

Programs

Maple

T := (n, k) -> binomial(n, k)^2 * binomial(2*k, k) / (k + 1):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Formula

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

T(n, k) = [x^n] hypergeom([1/2, -n, -n], [1, 2], 4*x).