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 A346505 #11 Sep 26 2023 06:47:35
%S A346505 1,2,4,12,44,172,700,2940,12652,55500,247260,1115740,5088908,23423020,
%T A346505 108659324,507520316,2384733868,11264884876,53464215580,254822253852,
%U A346505 1219182031820,5853309920748,28190437248700,136160853462524,659401832797676,3201141695492172,15575294057678428
%N A346505 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^2) / (1 - x + 2 * x^2).
%F A346505 a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=2..n-1} a(k) * a(n-k-1).
%F A346505 From _Vaclav Kotesovec_, Sep 26 2023: (Start)
%F A346505 G.f.: (1 - x*(1 - 2*x))*(1 - sqrt(1 - 4*x/(-1 + x - 2*x^2)^2))/(2*x).
%F A346505 a(n) ~ sqrt((69 + 57*sqrt(114) + 23*3^(5/6)*sqrt(38)*(9 + 2*sqrt(114))^(1/3) - 36*3^(1/6)*sqrt(38)*(9 + 2*sqrt(114))^(2/3) + 291*(27 + 6*sqrt(114))^(1/3) - 54*(27 + 6*sqrt(114))^(2/3))/(-72 - 16*sqrt(114) + 3^(11/6)*sqrt(38)*(9 + 2*sqrt(114))^(1/3) + 3^(1/6)*sqrt(38)*(9 + 2*sqrt(114))^(2/3) + 26*(27 + 6*sqrt(114))^(1/3) - 6*(27 + 6*sqrt(114))^(2/3))) * 2^(n - 1/2) * 3^(1/6 + 4*n/3) * ((9 + 2*sqrt(114))^((1/3)*(n-1)) / (sqrt(Pi) * n^(3/2) * (-15 + (27 + 6*sqrt(114))^(2/3))^n)). (End)
%t A346505 nmax = 26; A[_] = 0; Do[A[x_] = (1 + x A[x]^2)/(1 - x + 2 x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A346505 a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[a[k] a[n - k - 1], {k, 2, n - 1}]; Table[a[n], {n, 0, 26}]
%t A346505 CoefficientList[Series[(1 - x*(1 - 2*x)) * (1 - Sqrt[1 - 4*x/(-1 + x - 2*x^2)^2]) / (2*x), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Sep 26 2023 *)
%Y A346505 Cf. A004148, A006318, A047891, A078482, A082582, A346506.
%K A346505 nonn
%O A346505 0,2
%A A346505 _Ilya Gutkovskiy_, Jul 21 2021