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 A307884 #41 May 13 2021 02:36:05
%S A307884 1,1,1,1,0,1,1,-1,-2,1,1,-2,-3,0,1,1,-3,-2,11,6,1,1,-4,1,28,1,0,1,1,
%T A307884 -5,6,45,-74,-81,-20,1,1,-6,13,56,-255,-92,141,0,1,1,-7,22,55,-554,
%U A307884 477,1324,363,70,1,1,-8,33,36,-959,2376,2689,-3656,-1791,0,1
%N A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2).
%C A307884 Column k is the diagonal of the rational function 1 / ((1-x)*(1-y) + k*x*y). - _Seiichi Manyama_, Jul 11 2020
%C A307884 More generally, column k is the diagonal of the rational function r / ((1-r*x)*(1-r*y) + r-1 + (k-r+1)*r*x*y) for any nonzero real number r. - _Seiichi Manyama_, Jul 22 2020
%H A307884 Seiichi Manyama, <a href="/A307884/b307884.txt">Antidiagonals n = 0..139, flattened</a>
%F A307884 T(n,k) is the coefficient of x^n in the expansion of (1 - (k-1)*x - k*x^2)^n.
%F A307884 T(n,k) = Sum_{j=0..n} (-k)^j * binomial(n,j)^2.
%F A307884 T(n,k) = Sum_{j=0..n} (-k-1)^(n-j) * binomial(n,j) * binomial(n+j,j).
%F A307884 n * T(n,k) = -(k-1) * (2*n-1) * T(n-1,k) - (k+1)^2 * (n-1) * T(n-2,k).
%e A307884 Square array begins:
%e A307884 1, 1, 1, 1, 1, 1, 1, ...
%e A307884 1, 0, -1, -2, -3, -4, -5, ...
%e A307884 1, -2, -3, -2, 1, 6, 13, ...
%e A307884 1, 0, 11, 28, 45, 56, 55, ...
%e A307884 1, 6, 1, -74, -255, -554, -959, ...
%e A307884 1, 0, -81, -92, 477, 2376, 6475, ...
%e A307884 1, -20, 141, 1324, 2689, -804, -20195, ...
%t A307884 T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 13 2021 *)
%Y A307884 Columns k=2..4 give (-1)^n * A098332, A116091, (-1)^n * A098341.
%Y A307884 Main diagonal gives A307885.
%Y A307884 T(n,n-1) gives A335310.
%Y A307884 Cf. A307883, A336179.
%K A307884 sign,tabl
%O A307884 0,9
%A A307884 _Seiichi Manyama_, May 02 2019