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A367022 Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = 4^n * hypergeom([1/2, -n - 1, -n], [2, 2], x).

%I A367022 #18 Nov 22 2023 03:57:31
%S A367022 1,4,1,16,12,2,64,96,48,5,256,640,640,200,14,1024,3840,6400,4000,840,
%T A367022 42,4096,21504,53760,56000,23520,3528,132,16384,114688,401408,627200,
%U A367022 439040,131712,14784,429,65536,589824,2752512,6021120,6322176,3161088,709632,61776,1430
%N A367022 Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = 4^n * hypergeom([1/2, -n - 1, -n], [2, 2], x).
%F A367022 From _Detlef Meya_, Nov 20 2023: (Start)
%F A367022 T(n, k) = 4^(n - k)*binomial(n, k)*binomial(n+1, k)*binomial(2*k, k)/(k + 1)^2.
%F A367022 T(n, k) = A001263(n+1, k+1)*4^(n - k)*binomial(2*k, k)/(k + 1). (End)
%e A367022 Triangle T(n, k) starts:
%e A367022   [0]     1;
%e A367022   [1]     4,      1;
%e A367022   [2]    16,     12,       2;
%e A367022   [3]    64,     96,      48,       5;
%e A367022   [4]   256,    640,     640,     200,      14;
%e A367022   [5]  1024,   3840,    6400,    4000,     840,      42;
%e A367022   [6]  4096,  21504,   53760,   56000,   23520,    3528,    132;
%e A367022   [7] 16384, 114688,  401408,  627200,  439040,  131712,  14784,   429;
%e A367022   [8] 65536, 589824, 2752512, 6021120, 6322176, 3161088, 709632, 61776, 1430;
%p A367022 p := n -> 4^n*hypergeom([1/2, -n - 1, -n], [2, 2], x):
%p A367022 T := (n, k) -> coeff(simplify(p(n)), x, k):
%p A367022 seq(seq(T(n, k), k = 0..n), n = 0..8);
%t A367022 T[n_,k_]:=4^(n-k)*Binomial[n,k]*Binomial[n+1,k]*Binomial[2*k,k]/(k+1)^2;Flatten[Table[T[n,k],{n,0,8},{k,0,n}]] (* _Detlef Meya_, Nov 20 2023 *)
%Y A367022 Cf. A038845 (column 1), A128088, A005802, A246513, A001263.
%K A367022 nonn,tabl
%O A367022 0,2
%A A367022 _Peter Luschny_, Nov 06 2023