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 A307819 #45 May 07 2019 07:03:54
%S A307819 1,1,0,1,-1,0,1,-2,-1,0,1,-3,0,5,0,1,-4,3,16,-5,0,1,-5,8,27,-56,-11,0,
%T A307819 1,-6,15,32,-189,48,41,0,1,-7,24,25,-416,567,384,-29,0,1,-8,35,0,-725,
%U A307819 2176,189,-1920,-125,0,1,-9,48,-49,-1080,5625,-4864,-11259,3168,365,0
%N A307819 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2*k*x + k*(k+4)*x^2).
%H A307819 Seiichi Manyama, <a href="/A307819/b307819.txt">Antidiagonals n = 0..139, flattened</a>
%F A307819 A(n,k) is the coefficient of x^n in the expansion of (1 - k*x - k*x^2)^n.
%F A307819 A(n,k) = Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n,2*j) * binomial(2*j,j).
%F A307819 n * A(n,k) = -k * (2*n-1) * A(n-1,k) - k * (k+4) * (n-1) * A(n-2,k).
%e A307819 Square array begins:
%e A307819 1, 1, 1, 1, 1, 1, 1, ...
%e A307819 0, -1, -2, -3, -4, -5, -6, ...
%e A307819 0, -1, 0, 3, 8, 15, 24, ...
%e A307819 0, 5, 16, 27, 32, 25, 0, ...
%e A307819 0, -11, 48, 567, 2176, 5625, 11664, ...
%e A307819 0, 41, 384, 189, -4864, -24375, -74304, ...
%e A307819 0, -29, -1920, -11259, -23552, 9375, 228096, ...
%t A307819 A[n_, k_] := (-k)^n*Hypergeometric2F1[(1-n)/2, -n/2, 1, -4/k]; A[0, _] = 1; A[_, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, May 07 2019 *)
%Y A307819 Columns k=0..3 give A000007, (-1)^n * A098331, A116093, (-1)^n * A098340.
%Y A307819 Main diagonal gives A307911.
%Y A307819 Cf. A307860, A307884, A307910.
%K A307819 sign,tabl
%O A307819 0,8
%A A307819 _Seiichi Manyama_, May 05 2019