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 A307968 #23 May 12 2021 03:56:09
%S A307968 1,1,0,1,-1,0,1,-2,0,0,1,-3,2,2,0,1,-4,6,4,-3,0,1,-5,12,0,-24,-1,0,1,
%T A307968 -6,20,-16,-63,48,11,0,1,-7,30,-50,-96,297,24,-15,0,1,-8,42,-108,-75,
%U A307968 896,-621,-464,-13,0,1,-9,56,-196,72,1875,-3904,-1053,1376,77,0
%N A307968 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 + k*x + sqrt(1 + 2*k*x + k*(k+4)*x^2)).
%H A307968 Seiichi Manyama, <a href="/A307968/b307968.txt">Antidiagonals n = 0..139, flattened</a>
%F A307968 A(n,k) is the coefficient of x^n in the expansion of 1/(n+1) * (1 - k*x - k*x^2)^(n+1).
%F A307968 A(n,k) = Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n,2*j) * A000108(j).
%F A307968 (n+2) * A(n,k) = -k * (2*n+1) * A(n-1,k) - k * (k+4) * (n-1) * A(n-2,k).
%e A307968 Square array begins:
%e A307968 1, 1, 1, 1, 1, 1, 1, ...
%e A307968 0, -1, -2, -3, -4, -5, -6, ...
%e A307968 0, 0, 2, 6, 12, 20, 30, ...
%e A307968 0, 2, 4, 0, -16, -50, -108, ...
%e A307968 0, -3, -24, -63, -96, -75, 72, ...
%e A307968 0, -1, 48, 297, 896, 1875, 3024, ...
%e A307968 0, 11, 24, -621, -3904, -13125, -32184, ...
%e A307968 0, -15, -464, -1053, 6912, 53125, 200880, ...
%t A307968 T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-j)] * Binomial[n, 2*j] * CatalanNumber[j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 12 2021 *)
%Y A307968 Columns k=0..2 give A000007, A007440(n+1), A307969.
%Y A307968 Main diagonal gives A307946.
%Y A307968 Cf. A107267, A307819.
%K A307968 sign,tabl
%O A307968 0,8
%A A307968 _Seiichi Manyama_, May 08 2019