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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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).

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%I A367023 #14 Nov 22 2023 12:19:27
%S A367023 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,
%T A367023 1470,882,132,1,28,392,2450,6860,8232,3696,429,1,36,672,5880,24696,
%U A367023 49392,44352,15444,1430,1,45,1080,12600,74088,222264,332640,231660,64350,4862
%N 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).
%F A367023 T(2*n, n) = Sum_{k=0..n} CatalanNumber(n)^2 * binomial(n + k, k).
%F A367023 From _Detlef Meya_, Nov 22 2023: (Start)
%F A367023 T(n, k) = binomial(n, k)*binomial(n+1, k)*binomial(2*k, k)/(k+1)^2.
%F A367023 T(n, k) = A001263(n+1, k+1)*binomial(2*k, k)/(k+1). (End)
%e A367023 Triangle T(n, k) starts:
%e A367023   [0] 1;
%e A367023   [1] 1,  1;
%e A367023   [2] 1,  3,    2;
%e A367023   [3] 1,  6,   12,     5;
%e A367023   [4] 1, 10,   40,    50,    14;
%e A367023   [5] 1, 15,  100,   250,   210,     42;
%e A367023   [6] 1, 21,  210,   875,  1470,    882,    132;
%e A367023   [7] 1, 28,  392,  2450,  6860,   8232,   3696,    429;
%e A367023   [8] 1, 36,  672,  5880, 24696,  49392,  44352,  15444,  1430;
%e A367023   [9] 1, 45, 1080, 12600, 74088, 222264, 332640, 231660, 64350, 4862;
%p A367023 p := n -> hypergeom([1/2, -n - 1, -n], [2, 2], 4*x):
%p A367023 T := (n, k) -> coeff(simplify(p(n)), x, k):
%p A367023 seq(seq(T(n, k), k = 0..n), n = 0..9);
%t A367023 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}]]
%t A367023 (* _Detlef Meya_, Nov 22 2023 *)
%Y A367023 Cf. A128088 (row sums), A358368 (central terms), A367022.
%K A367023 nonn,tabl
%O A367023 0,5
%A A367023 _Peter Luschny_, Nov 06 2023