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 A307847 #35 May 13 2021 02:35:40
%S A307847 1,1,1,1,1,1,1,1,3,1,1,1,9,7,1,1,1,19,25,19,1,1,1,33,55,145,51,1,1,1,
%T A307847 51,97,595,561,141,1,1,1,73,151,1729,2611,2841,393,1,1,1,99,217,4051,
%U A307847 8001,22141,12489,1107,1,1,1,129,295,8209,19251,105441,119449,60705,3139,1
%N A307847 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1-4*k^2)*x^2).
%H A307847 Seiichi Manyama, <a href="/A307847/b307847.txt">Antidiagonals n = 0..139, flattened</a>
%F A307847 A(n,k) is the constant term in the expansion of (k/x + 1 + k*x)^n.
%F A307847 A(n,k) is the coefficient of x^n in the expansion of (1 + x + (k*x)^2)^n.
%F A307847 A(n,k) = Sum_{j=0..n} (1-2*k)^(n-j) * k^j * binomial(n,j) * binomial(2*j,j).
%F A307847 A(n,k) = Sum_{j=0..n} (1+2*k)^(n-j) * (-k)^j * binomial(n,j) * binomial(2*j,j).
%F A307847 A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * binomial(n,2*j) * binomial(2*j,j).
%F A307847 n * A(n,k) = (2*n-1) * A(n-1,k) - (1-4*k^2) * (n-1) * A(n-2,k).
%e A307847 Square array begins:
%e A307847 1, 1, 1, 1, 1, 1, ...
%e A307847 1, 1, 1, 1, 1, 1, ...
%e A307847 1, 3, 9, 19, 33, 51, ...
%e A307847 1, 7, 25, 55, 97, 151, ...
%e A307847 1, 19, 145, 595, 1729, 4051, ...
%e A307847 1, 51, 561, 2611, 8001, 19251, ...
%e A307847 1, 141, 2841, 22141, 105441, 369501, ...
%e A307847 1, 393, 12489, 119449, 627873, 2319801, ...
%t A307847 T[n_, k_] := Sum[If[k == j == 0, 1, k^(2*j)] * Binomial[n, 2*j] * Binomial[2*j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 13 2021 *)
%Y A307847 Columns k=0..2 give A000012, A002426, A084605.
%Y A307847 Main diagonal gives A307844.
%Y A307847 Cf. A292627, A307855.
%K A307847 nonn,tabl
%O A307847 0,9
%A A307847 _Seiichi Manyama_, May 01 2019