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 A353596 #17 Oct 04 2023 04:49:14
%S A353596 1,0,2,2,0,-2,0,-4,0,4,-2,0,12,0,-10,0,12,0,-40,0,28,4,0,-60,0,140,0,
%T A353596 -84,0,-40,0,280,0,-504,0,264,-10,0,280,0,-1260,0,1848,0,-858,0,140,0,
%U A353596 -1680,0,5544,0,-6864,0,2860,28,0,-1260,0,9240,0,-24024,0,25740,0,-9724
%N A353596 Triangle read by rows, T(n, k) = [x^k] (-2)^n*GegenbauerC(n, -1/2, x).
%e A353596 Triangle T(n, k) starts:
%e A353596 [0] 1;
%e A353596 [1] 0, 2;
%e A353596 [2] 2, 0, -2;
%e A353596 [3] 0, -4, 0, 4;
%e A353596 [4] -2, 0, 12, 0, -10;
%e A353596 [5] 0, 12, 0, -40, 0, 28;
%e A353596 [6] 4, 0, -60, 0, 140, 0, -84;
%e A353596 [7] 0, -40, 0, 280, 0, -504, 0, 264;
%e A353596 [8] -10, 0, 280, 0, -1260, 0, 1848, 0, -858;
%e A353596 [9] 0, 140, 0, -1680, 0, 5544, 0, -6864, 0, 2860;
%e A353596 .
%e A353596 Unsigned antidiagonals |T(n+k, n-k)|:
%e A353596 [0] 1;
%e A353596 [1] 2, 2;
%e A353596 [2] 2, 4, 2;
%e A353596 [3] 4, 12, 12, 4;
%e A353596 [4] 10, 40, 60, 40, 10;
%e A353596 [5] 28, 140, 280, 280, 140, 28;
%e A353596 [6] 84, 504, 1260, 1680, 1260, 504, 84;
%p A353596 g := n -> (-2)^n*GegenbauerC(n, -1/2, x):
%p A353596 seq(print(seq(coeff(simplify(g(n)), x, k), k = 0..n)), n = 0..9);
%t A353596 s={}; For[n=0,n<11,n++,For[k=0,k<n+1,k++,If[EvenQ[n+k],If[Mod[n+k,4]==0,AppendTo[s,Binomial[n+k,(n+k)/2]/(1-(n+k))*Binomial[(n+k)/2,k]],AppendTo[s,(-1)*Binomial[n+k,(n+k)/2]/(1-(n+k))*Binomial[(n+k)/2,k]]],AppendTo[s,0]]]]; s (* _Detlef Meya_, Oct 03 2023 *)
%Y A353596 Diagonals (also divided by 2^k): A002420 (main), A028329 (main-2) (also A000984), A005430 (main-4) (also A002457), A002802 (main-6).
%K A353596 sign,tabl
%O A353596 0,3
%A A353596 _Peter Luschny_, May 06 2022