cp's OEIS Frontend

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

%I A367178 #7 Nov 07 2023 14:43:04
%S A367178 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 A367178 3150,1512,132,1,49,882,6125,17150,18522,6468,429,1,64,1568,15680,
%U A367178 68600,131712,103488,27456,1430,1,81,2592,35280,222264,666792,931392,555984,115830,4862
%N A367178 Triangle read by rows. T(n, k) = binomial(n, k)^2 * CatalanNumber(k).
%F A367178 T(n, k) = binomial(n, k)^2 * binomial(2*k, k) / (k + 1).
%F A367178 T(n, k) = [x^n] hypergeom([1/2, -n, -n], [1, 2], 4*x).
%e A367178 Triangle T(n, k) starts:
%e A367178   [0] 1;
%e A367178   [1] 1,  1;
%e A367178   [2] 1,  4,    2;
%e A367178   [3] 1,  9,   18,     5;
%e A367178   [4] 1, 16,   72,    80,     14;
%e A367178   [5] 1, 25,  200,   500,    350,     42;
%e A367178   [6] 1, 36,  450,  2000,   3150,   1512,    132;
%e A367178   [7] 1, 49,  882,  6125,  17150,  18522,   6468,    429;
%e A367178   [8] 1, 64, 1568, 15680,  68600, 131712, 103488,  27456,   1430;
%e A367178   [9] 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862;
%p A367178 T := (n, k) -> binomial(n, k)^2 * binomial(2*k, k) / (k + 1):
%p A367178 seq(seq(T(n, k), k = 0..n), n = 0..9);
%Y A367178 Cf. A086618 (row sums), A186415 (central column), A000108 (main diagonal).
%Y A367178 Cf. A098474, A367022, A367023, A387024, A387024, A367177.
%K A367178 nonn,tabl
%O A367178 0,5
%A A367178 _Peter Luschny_, Nov 07 2023