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 A349013 #7 Nov 06 2021 09:07:16
%S A349013 1,1,3,14,119,1938,62291,3990822,510954584,130812494690,
%T A349013 66977027614263,68584738950054184,140461679588587238360,
%U A349013 575331176899085656611498,4713113282218244019907299347,77219649166807400596911172015640,2530333473325322974256848862067093128
%N A349013 G.f. A(x) satisfies: A(x) = (1 + x * A(2*x)) / (1 - x^2 * A(x)).
%F A349013 a(0) = 1; a(n) = 2^(n-1) * a(n-1) + Sum_{k=0..n-2} a(k) * a(n-k-2).
%F A349013 a(n) ~ c * 2^(n*(n-1)/2), where c = 1.903611332661849505687584024897203124045921097899485586512432930506328189... - _Vaclav Kotesovec_, Nov 06 2021
%t A349013 nmax = 16; A[_] = 0; Do[A[x_] = (1 + x A[2 x])/(1 - x^2 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349013 a[0] = 1; a[n_] := a[n] = 2^(n - 1) a[n - 1] + Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 16}]
%Y A349013 Cf. A000108, A001006, A135867, A348878, A349012.
%K A349013 nonn
%O A349013 0,3
%A A349013 _Ilya Gutkovskiy_, Nov 05 2021