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 A349255 #15 Aug 01 2023 09:33:46
%S A349255 1,1,7,47,369,3113,27631,254239,2403361,23201393,227771831,2266983119,
%T A349255 22822484497,231994748633,2377894546783,24548520253247,
%U A349255 255026759000897,2664111200687969,27967731861910759,294900120348032623,3121862973452544433,33167268461833410569
%N A349255 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - 2 * x * A(x)^2)).
%H A349255 Seiichi Manyama, <a href="/A349255/b349255.txt">Table of n, a(n) for n = 0..950</a>
%F A349255 a(n) = (-1)^n + 2 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
%F A349255 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+k,n-k) * 2^k * binomial(3*k,k) / (2*k+1).
%F A349255 a(n) = (-1)^n*hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], (3/2)^3). - _Peter Luschny_, Nov 12 2021
%F A349255 a(n) ~ sqrt(171 + 23*sqrt(57)) * (23 + 3*sqrt(57))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(2*n + 5/2)). - _Vaclav Kotesovec_, Nov 13 2021
%t A349255 nmax = 21; A[_] = 0; Do[A[x_] = 1/((1 + x) (1 - 2 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349255 a[n_] := a[n] = (-1)^n + 2 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 21}]
%t A349255 Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] 2^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 21}]
%t A349255 a[n_] := (-1)^n*HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, (3/2)^3]; Table[a[n], {n, 0, 21}] (* _Peter Luschny_, Nov 12 2021 *)
%Y A349255 Cf. A001003, A001764, A153231, A337168, A346627, A349253, A349254, A349256.
%K A349255 nonn
%O A349255 0,3
%A A349255 _Ilya Gutkovskiy_, Nov 12 2021