A367023 Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = hypergeom([1/2, -n - 1, -n], [2, 2], 4*x).
1, 1, 1, 1, 3, 2, 1, 6, 12, 5, 1, 10, 40, 50, 14, 1, 15, 100, 250, 210, 42, 1, 21, 210, 875, 1470, 882, 132, 1, 28, 392, 2450, 6860, 8232, 3696, 429, 1, 36, 672, 5880, 24696, 49392, 44352, 15444, 1430, 1, 45, 1080, 12600, 74088, 222264, 332640, 231660, 64350, 4862
Offset: 0
Examples
Triangle T(n, k) starts: [0] 1; [1] 1, 1; [2] 1, 3, 2; [3] 1, 6, 12, 5; [4] 1, 10, 40, 50, 14; [5] 1, 15, 100, 250, 210, 42; [6] 1, 21, 210, 875, 1470, 882, 132; [7] 1, 28, 392, 2450, 6860, 8232, 3696, 429; [8] 1, 36, 672, 5880, 24696, 49392, 44352, 15444, 1430; [9] 1, 45, 1080, 12600, 74088, 222264, 332640, 231660, 64350, 4862;
Programs
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Maple
p := n -> hypergeom([1/2, -n - 1, -n], [2, 2], 4*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,k]*Binomial[n+1,k]*Binomial[2*k,k]/(k+1)^2;Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Nov 22 2023 *)
Formula
T(2*n, n) = Sum_{k=0..n} CatalanNumber(n)^2 * binomial(n + k, k).
From Detlef Meya, Nov 22 2023: (Start)
T(n, k) = binomial(n, k)*binomial(n+1, k)*binomial(2*k, k)/(k+1)^2.
T(n, k) = A001263(n+1, k+1)*binomial(2*k, k)/(k+1). (End)