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).
1, 4, 1, 16, 12, 2, 64, 96, 48, 5, 256, 640, 640, 200, 14, 1024, 3840, 6400, 4000, 840, 42, 4096, 21504, 53760, 56000, 23520, 3528, 132, 16384, 114688, 401408, 627200, 439040, 131712, 14784, 429, 65536, 589824, 2752512, 6021120, 6322176, 3161088, 709632, 61776, 1430
Offset: 0
Examples
Triangle T(n, k) starts: [0] 1; [1] 4, 1; [2] 16, 12, 2; [3] 64, 96, 48, 5; [4] 256, 640, 640, 200, 14; [5] 1024, 3840, 6400, 4000, 840, 42; [6] 4096, 21504, 53760, 56000, 23520, 3528, 132; [7] 16384, 114688, 401408, 627200, 439040, 131712, 14784, 429; [8] 65536, 589824, 2752512, 6021120, 6322176, 3161088, 709632, 61776, 1430;
Programs
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Maple
p := n -> 4^n*hypergeom([1/2, -n - 1, -n], [2, 2], x): T := (n, k) -> coeff(simplify(p(n)), x, k): seq(seq(T(n, k), k = 0..n), n = 0..8);
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Mathematica
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 *)
Formula
From Detlef Meya, Nov 20 2023: (Start)
T(n, k) = 4^(n - k)*binomial(n, k)*binomial(n+1, k)*binomial(2*k, k)/(k + 1)^2.
T(n, k) = A001263(n+1, k+1)*4^(n - k)*binomial(2*k, k)/(k + 1). (End)