A367025 Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = (1 - hypergeom([-1/2, -n - 1, -n - 1], [1, 1], 4*x)) / (2*x).
1, 4, 1, 9, 9, 2, 16, 36, 32, 5, 25, 100, 200, 125, 14, 36, 225, 800, 1125, 504, 42, 49, 441, 2450, 6125, 6174, 2058, 132, 64, 784, 6272, 24500, 43904, 32928, 8448, 429, 81, 1296, 14112, 79380, 222264, 296352, 171072, 34749, 1430
Offset: 0
Examples
Triangle T(n, k) starts: [0] 1; [1] 4, 1; [2] 9, 9, 2; [3] 16, 36, 32, 5; [4] 25, 100, 200, 125, 14; [5] 36, 225, 800, 1125, 504, 42; [6] 49, 441, 2450, 6125, 6174, 2058, 132; [7] 64, 784, 6272, 24500, 43904, 32928, 8448, 429; [8] 81, 1296, 14112, 79380, 222264, 296352, 171072, 34749, 1430; [9] 100, 2025, 28800, 220500, 889056, 1852200, 1900800, 868725, 143000, 4862;
Programs
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Maple
p := n -> (1 - hypergeom([-1/2, -n-1, -n-1], [1, 1], 4*x)) / (2*x): T := (n, k) -> coeff(simplify(p(n)), x, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
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Mathematica
T[n_,k_]:=Binomial[n+1,n-k]^2*Binomial[2*k,k]/(k+1);Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Nov 19 2023 *)
Formula
T(n,k) = binomial(n+1,n-k)^2*binomial(2*k,k)/(k+1). - Detlef Meya, Nov 19 2023