cp's OEIS Frontend

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.

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).

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%I A367025 #14 Nov 20 2023 11:52:33
%S A367025 1,4,1,9,9,2,16,36,32,5,25,100,200,125,14,36,225,800,1125,504,42,49,
%T A367025 441,2450,6125,6174,2058,132,64,784,6272,24500,43904,32928,8448,429,
%U A367025 81,1296,14112,79380,222264,296352,171072,34749,1430
%N 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).
%F A367025 T(n,k) = binomial(n+1,n-k)^2*binomial(2*k,k)/(k+1). - _Detlef Meya_, Nov 19 2023
%e A367025 Triangle T(n, k) starts:
%e A367025   [0]   1;
%e A367025   [1]   4,    1;
%e A367025   [2]   9,    9,     2;
%e A367025   [3]  16,   36,    32,      5;
%e A367025   [4]  25,  100,   200,    125,     14;
%e A367025   [5]  36,  225,   800,   1125,    504,      42;
%e A367025   [6]  49,  441,  2450,   6125,   6174,    2058,     132;
%e A367025   [7]  64,  784,  6272,  24500,  43904,   32928,    8448,    429;
%e A367025   [8]  81, 1296, 14112,  79380, 222264,  296352,  171072,  34749,   1430;
%e A367025   [9] 100, 2025, 28800, 220500, 889056, 1852200, 1900800, 868725, 143000, 4862;
%p A367025 p := n -> (1 - hypergeom([-1/2, -n-1, -n-1], [1, 1], 4*x)) / (2*x):
%p A367025 T := (n, k) -> coeff(simplify(p(n)), x, k):
%p A367025 seq(seq(T(n, k), k = 0..n), n = 0..9);
%t A367025 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 *)
%Y A367025 Cf. A000290 (first column), A000108 (main diagonal).
%Y A367025 Cf. A367022, A367023, A387024.
%K A367025 nonn,tabl
%O A367025 0,2
%A A367025 _Peter Luschny_, Nov 07 2023