%I #7 Nov 07 2023 14:43:04

%S 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,

%T 3150,1512,132,1,49,882,6125,17150,18522,6468,429,1,64,1568,15680,

%U 68600,131712,103488,27456,1430,1,81,2592,35280,222264,666792,931392,555984,115830,4862

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

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

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

%e Triangle T(n, k) starts:

%e [0] 1;

%e [1] 1, 1;

%e [2] 1, 4, 2;

%e [3] 1, 9, 18, 5;

%e [4] 1, 16, 72, 80, 14;

%e [5] 1, 25, 200, 500, 350, 42;

%e [6] 1, 36, 450, 2000, 3150, 1512, 132;

%e [7] 1, 49, 882, 6125, 17150, 18522, 6468, 429;

%e [8] 1, 64, 1568, 15680, 68600, 131712, 103488, 27456, 1430;

%e [9] 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862;

%p T := (n, k) -> binomial(n, k)^2 * binomial(2*k, k) / (k + 1):

%p seq(seq(T(n, k), k = 0..n), n = 0..9);

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

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

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Nov 07 2023