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 A333989 #17 Sep 04 2020 15:20:26
%S A333989 1,1,1,1,0,1,1,-1,-4,1,1,-2,-7,0,1,1,-3,-8,23,16,1,1,-4,-7,64,17,0,1,
%T A333989 1,-5,-4,117,-128,-241,-64,1,1,-6,1,176,-527,-512,329,0,1,1,-7,8,235,
%U A333989 -1264,237,4096,1511,256,1,1,-8,17,288,-2399,3776,11753,-8192,-5983,0,1
%N A333989 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1+(k-1)*x) / (1+2*(k-1)*x+((k+1)*x)^2).
%F A333989 T(n,k) = Sum_{j=0..n} (-k)^j * binomial(2*n,2*j).
%F A333989 T(0,k) = 1, T(1,k) = 1-k and T(n,k) = -2 * (k-1) * T(n-1,k) - (k+1)^2 * T(n-2,k) for n>1.
%e A333989 Square array begins:
%e A333989 1, 1, 1, 1, 1, 1, ...
%e A333989 1, 0, -1, -2, -3, -4, ...
%e A333989 1, -4, -7, -8, -7, -4, ...
%e A333989 1, 0, 23, 64, 117, 176, ...
%e A333989 1, 16, 17, -128, -527, -1264, ...
%e A333989 1, 0, -241, -512, 237, 3776, ...
%t A333989 T[n_, 0] := 1; T[n_, k_] := Sum[(-k)^j * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Sep 04 2020 *)
%o A333989 (PARI) {T(n, k) = sum(j=0, n, (-k)^j*binomial(2*n, 2*j))}
%Y A333989 Main diagonal gives A333991.
%Y A333989 Cf. A307884, A333988.
%K A333989 sign,tabl
%O A333989 0,9
%A A333989 _Seiichi Manyama_, Sep 04 2020