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 A171213 #5 Feb 07 2025 05:53:16
%S A171213 1,3,54,3402,618921,318392208,474852630879,2094575471899362,
%T A171213 27570620677894020891,1086589159409074932937593,
%U A171213 128377839432663886566934695744,45490432595875817814676362194769627
%N A171213 G.f.: A(x) satisfies A(x) = x + x*A(A(3*x)).
%C A171213 More generally, if F(x) = x + x*F(F(qx)), then
%C A171213 F(x) = x + x*F(qx) + x*F(qx)*F(qF(qx)) + x*F(qx)*F(qF(qx))*F(qF(qF(qx))) +...
%C A171213 with a simple solution at q=1/2:
%C A171213 F(x) = x/(1-x/2) satisfies: F(x) = x + x*F(F(x/2)).
%e A171213 G.f.: A(x) = x + 3*x^2 + 54*x^3 + 3402*x^4 + 618921*x^5 +...
%e A171213 A(A(x)) = x + 6*x^2 + 126*x^3 + 7641*x^4 + 1310256*x^5 +...+ a(n)*x^n/3^(n-1) +...
%e A171213 As a formal series involving products of iterations of the g.f.,
%e A171213 A(x) = x + x*A(3x) + x*A(3x)*A(3A(3x)) + x*A(3x)*A(3A(3x))*A(3A(3A(3x))) +...
%e A171213 which, upon replacing x with A(3x), yields:
%e A171213 A(A(3x)) = A(3x) + A(3x)*A(3A(3x)) + A(3x)*A(3A(3x))*A(3A(3A(3x))) +...
%e A171213 thus A(x) = x + x*A(A(3x)).
%o A171213 (PARI) {a(n,q=3)=local(A=x+x^2);for(i=1,n,A=x+x*subst(A,x,subst(A,x,q*x+O(x^n))));polcoeff(A,n)}
%Y A171213 Cf. A171212 (q=2), A171214 (q=1/3).
%K A171213 nonn
%O A171213 1,2
%A A171213 _Paul D. Hanna_, Dec 08 2009