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 A337369 #30 Aug 31 2020 04:24:16
%S A337369 1,1,6,1,7,30,1,8,51,140,1,9,74,393,630,1,10,99,736,3139,2772,1,11,
%T A337369 126,1175,7606,25653,12012,1,12,155,1716,14499,80464,212941,51480,1,
%U A337369 13,186,2365,24310,183195,864772,1787607,218790,1,14,219,3128,37555,352716,2351805,9400192,15134931,923780
%N A337369 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of sqrt(2 / ( (1-2*(k+4)*x+((k-4)*x)^2) * (1+(k-4)*x+sqrt(1-2*(k+4)*x+((k-4)*x)^2)) )).
%H A337369 Seiichi Manyama, <a href="/A337369/b337369.txt">Antidiagonals n = 0..139, flattened</a>
%F A337369 T(n,k) = Sum_{j=0..n} k^(n-j) * binomial(2*j,j) * binomial(2*n+1,2*j).
%F A337369 T(0,k) = 1, T(1,k) = k+6 and n * (2*n+1) * (4*n-3) * T(n,k) = (4*n-1) * (4*(k+4)*n^2-2*(k+4)*n-k-2) * T(n-1,k) - (k-4)^2 * (n-1) * (2*n-1) * (4*n+1) * T(n-2,k) for n > 1. - _Seiichi Manyama_, Aug 29 2020
%F A337369 For fixed k > 0, T(n,k) ~ (2 + sqrt(k))^(2*n + 3/2) / sqrt(8*k*Pi*n). - _Vaclav Kotesovec_, Aug 31 2020
%e A337369 Square array begins:
%e A337369 1, 1, 1, 1, 1, 1, ...
%e A337369 6, 7, 8, 9, 10, 11, ...
%e A337369 30, 51, 74, 99, 126, 155, ...
%e A337369 140, 393, 736, 1175, 1716, 2365, ...
%e A337369 630, 3139, 7606, 14499, 24310, 37555, ...
%e A337369 2772, 25653, 80464, 183195, 352716, 610897, ...
%t A337369 T[n_, k_] := Sum[If[k == 0, Boole[n == j], k^(n - j)] * Binomial[2*j, j] * Binomial[2*n + 1, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Aug 25 2020 *)
%o A337369 (PARI) {T(n, k) = sum(j=0, n, k^(n-j)*binomial(2*j, j)*binomial(2*n+1, 2*j))}
%Y A337369 Columns k=0..5 give A002457, A273055, A337370, A245927, A002458, A243947.
%Y A337369 Main diagonal gives A337387.
%Y A337369 Cf. A337389, A337464.
%K A337369 nonn,tabl
%O A337369 0,3
%A A337369 _Seiichi Manyama_, Aug 25 2020