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A367024 Triangle read by rows, T(n, k) = [x^k] -hypergeom([-1/2, -n, -n], [1, 1], 4*x).

%I A367024 #9 Nov 07 2023 14:43:18
%S A367024 -1,-1,2,-1,8,2,-1,18,18,4,-1,32,72,64,10,-1,50,200,400,250,28,-1,72,
%T A367024 450,1600,2250,1008,84,-1,98,882,4900,12250,12348,4116,264,-1,128,
%U A367024 1568,12544,49000,87808,65856,16896,858,-1,162,2592,28224,158760,444528,592704,342144,69498,2860
%N A367024 Triangle read by rows, T(n, k) = [x^k] -hypergeom([-1/2, -n, -n], [1, 1], 4*x).
%F A367024 T(n, k) = binomial(n, k)^2 * binomial(2*k, k) / (2*k - 1).
%e A367024 Triangle T(n, k) starts:
%e A367024   [0] -1;
%e A367024   [1] -1,   2;
%e A367024   [2] -1,   8,    2;
%e A367024   [3] -1,  18,   18,     4;
%e A367024   [4] -1,  32,   72,    64,     10;
%e A367024   [5] -1,  50,  200,   400,    250,     28;
%e A367024   [6] -1,  72,  450,  1600,   2250,   1008,     84;
%e A367024   [7] -1,  98,  882,  4900,  12250,  12348,   4116,    264;
%e A367024   [8] -1, 128, 1568, 12544,  49000,  87808,  65856,  16896,   858;
%e A367024   [9] -1, 162, 2592, 28224, 158760, 444528, 592704, 342144, 69498, 2860;
%p A367024 p := n -> -hypergeom([-1/2, -n, -n], [1, 1], 4*x):
%p A367024 T := (n, k) -> coeff(simplify(p(n)), x, k):
%p A367024 seq(seq(T(n, k), k = 0..n), n = 0..9);
%Y A367024 Cf. A246065 (row sums), -A002420 and A284016 (main diagonal).
%Y A367024 Cf. A367022, A367023, A387025.
%K A367024 sign,tabl
%O A367024 0,3
%A A367024 _Peter Luschny_, Nov 07 2023