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.
%I A337419 #24 Aug 29 2020 02:18:43
%S A337419 1,1,2,1,1,6,1,0,-5,20,1,-1,-14,-41,70,1,-2,-21,-48,-125,252,1,-3,-26,
%T A337419 -7,198,131,924,1,-4,-29,76,739,2080,3301,3432,1,-5,-30,195,1222,1629,
%U A337419 1780,15625,12870,1,-6,-29,344,1395,-3772,-26859,-57120,16115,48620
%N A337419 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt((1-(k+4)*x+sqrt(1+2*(k-4)*x+((k+4)*x)^2)) / (2 * (1+2*(k-4)*x+((k+4)*x)^2))).
%H A337419 Seiichi Manyama, <a href="/A337419/b337419.txt">Antidiagonals n = 0..139, flattened</a>
%F A337419 T(n,k) = Sum_{j=0..n} (-k)^(n-j) * binomial(2*j,j) * binomial(2*n,2*j).
%F A337419 T(0,k) = 1, T(1,k) = 2-k and n * (2*n-1) * (4*n-5) * T(n,k) = (4*n-3) * (-4*(k-4)*n^2+6*(k-4)*n-k+6) * T(n-1,k) - (k+4)^2 * (n-1) * (2*n-3) * (4*n-1) * T(n-2,k) for n > 1. - _Seiichi Manyama_, Aug 28 2020
%e A337419 Square array begins:
%e A337419 1, 1, 1, 1, 1, 1, ...
%e A337419 2, 1, 0, -1, -2, -3, ...
%e A337419 6, -5, -14, -21, -26, -29, ...
%e A337419 20, -41, -48, -7, 76, 195, ...
%e A337419 70, -125, 198, 739, 1222, 1395, ...
%e A337419 252, 131, 2080, 1629, -3772, -14873, ...
%t A337419 T[n_, k_] := Sum[If[k == 0, Boole[n == j], (-k)^(n - j)] * Binomial[2*j, j] * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Aug 27 2020 *)
%o A337419 (PARI) {T(n, k) = sum(j=0, n, (-k)^(n-j)*binomial(2*j, j)*binomial(2*n, 2*j))}
%Y A337419 Columns k=0..4 give A000984, A337393, A337421, A337422, A337396.
%Y A337419 Main diagonal gives A337420.
%Y A337419 Cf. A337389, A337464.
%K A337419 sign,tabl
%O A337419 0,3
%A A337419 _Seiichi Manyama_, Aug 27 2020