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Triangle read by rows. T(n, k) = binomial(n + k, 2*k)^2.
1

%I #11 Feb 14 2024 05:42:12

%S 1,1,1,1,9,1,1,36,25,1,1,100,225,49,1,1,225,1225,784,81,1,1,441,4900,

%T 7056,2025,121,1,1,784,15876,44100,27225,4356,169,1,1,1296,44100,

%U 213444,245025,81796,8281,225,1,1,2025,108900,853776,1656369,1002001,207025,14400,289,1

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

%F T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n + k, 2*k) * Pochhammer(n - k + c, 2*k) * z^k / (2*k)! and c = 1.

%F T(n, k) = [z^k] hypergeom([-n, -n, 1 + n, 1 + n], [1/2, 1/2, 1], z/16).

%e Triangle starts:

%e [0] 1;

%e [1] 1, 1;

%e [2] 1, 9, 1;

%e [3] 1, 36, 25, 1;

%e [4] 1, 100, 225, 49, 1;

%e [5] 1, 225, 1225, 784, 81, 1;

%e [6] 1, 441, 4900, 7056, 2025, 121, 1;

%e [7] 1, 784, 15876, 44100, 27225, 4356, 169, 1;

%t Table[Binomial[n + k, 2*k]^2, {n, 0, 7}, {k, 0, n}] // Flatten

%Y Shifted bisection of A182878.

%Y Cf. A370233 (c=2), A188648 (row sums), A188662 (central terms).

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Feb 12 2024