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A367177 Triangle read by rows, T(n, k) = [x^k] hypergeom([1/2, -n, -n], [1, 1], 4*x).

%I A367177 #6 Nov 07 2023 14:43:09
%S A367177 1,1,2,1,8,6,1,18,54,20,1,32,216,320,70,1,50,600,2000,1750,252,1,72,
%T A367177 1350,8000,15750,9072,924,1,98,2646,24500,85750,111132,45276,3432,1,
%U A367177 128,4704,62720,343000,790272,724416,219648,12870
%N A367177 Triangle read by rows, T(n, k) = [x^k] hypergeom([1/2, -n, -n], [1, 1], 4*x).
%F A367177 T(n, k) = binomial(n, k)^2 * binomial(2*k, k).
%e A367177 Triangle T(n, k) starts:
%e A367177   [0] 1;
%e A367177   [1] 1,   2;
%e A367177   [2] 1,   8,    6;
%e A367177   [3] 1,  18,   54,     20;
%e A367177   [4] 1,  32,  216,    320,      70;
%e A367177   [5] 1,  50,  600,   2000,    1750,     252;
%e A367177   [6] 1,  72, 1350,   8000,   15750,    9072,     924;
%e A367177   [7] 1,  98, 2646,  24500,   85750,  111132,   45276,    3432;
%e A367177   [8] 1, 128, 4704,  62720,  343000,  790272,  724416,  219648,   12870;
%e A367177   [9] 1, 162, 7776, 141120, 1111320, 4000752, 6519744, 4447872, 1042470, 48620;
%p A367177 p := n -> hypergeom([1/2, -n, -n], [1, 1], 4*x):
%p A367177 T := (n, k) -> coeff(simplify(p(n)), x, k):
%p A367177 seq(seq(T(n, k), k = 0..n), n = 0..9);
%Y A367177 Cf. A002893 (row sum), A002897 (central column), A000984 (main diagonal).
%Y A367177 Cf. A367022, A367023, A387024.
%K A367177 nonn,tabl
%O A367177 0,3
%A A367177 _Peter Luschny_, Nov 07 2023