%I #6 Nov 07 2023 14:43:09
%S 1,1,2,1,8,6,1,18,54,20,1,32,216,320,70,1,50,600,2000,1750,252,1,72,
%T 1350,8000,15750,9072,924,1,98,2646,24500,85750,111132,45276,3432,1,
%U 128,4704,62720,343000,790272,724416,219648,12870
%N Triangle read by rows, T(n, k) = [x^k] hypergeom([1/2, -n, -n], [1, 1], 4*x).
%F T(n, k) = binomial(n, k)^2 * binomial(2*k, k).
%e Triangle T(n, k) starts:
%e [0] 1;
%e [1] 1, 2;
%e [2] 1, 8, 6;
%e [3] 1, 18, 54, 20;
%e [4] 1, 32, 216, 320, 70;
%e [5] 1, 50, 600, 2000, 1750, 252;
%e [6] 1, 72, 1350, 8000, 15750, 9072, 924;
%e [7] 1, 98, 2646, 24500, 85750, 111132, 45276, 3432;
%e [8] 1, 128, 4704, 62720, 343000, 790272, 724416, 219648, 12870;
%e [9] 1, 162, 7776, 141120, 1111320, 4000752, 6519744, 4447872, 1042470, 48620;
%p p := n -> hypergeom([1/2, -n, -n], [1, 1], 4*x):
%p T := (n, k) -> coeff(simplify(p(n)), x, k):
%p seq(seq(T(n, k), k = 0..n), n = 0..9);
%Y Cf. A002893 (row sum), A002897 (central column), A000984 (main diagonal).
%Y Cf. A367022, A367023, A387024.
%K nonn,tabl
%O 0,3
%A _Peter Luschny_, Nov 07 2023
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