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 A349318 #39 Nov 25 2021 02:58:59
%S A349318 1,1,5,28,171,1113,7590,53588,388519,2876003,21648065,165193576,
%T A349318 1275043280,9936953788,78087083456,618049278976,4922606097263,
%U A349318 39425205882007,317316076325015,2565216211152700,20819872339143179,169586043613302169,1385856599443533442
%N A349318 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - 2 * x).
%F A349318 a(0) = a(1) = 1; a(n) = 2 * a(n-1) + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
%F A349318 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(3*k,k) * 2^(n-k) / (2*k+1).
%F A349318 a(n) ~ 35^(n + 1/2) / (3 * sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). - _Vaclav Kotesovec_, Nov 25 2021
%t A349318 nmax = 22; A[_] = 0; Do[A[x_] = 1 + x A[x]^3/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349318 a[0] = a[1] = 1; a[n_] := a[n] = 2 a[n - 1] + 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, 22}]
%t A349318 Table[Sum[Binomial[n - 1, k - 1] Binomial[3 k, k] 2^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]
%Y A349318 Cf. A001764, A005572, A307678, A349531, A349532.
%K A349318 nonn
%O A349318 0,3
%A A349318 _Ilya Gutkovskiy_, Nov 21 2021