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 A171212 #6 Feb 07 2025 00:51:58
%S A171212 0,1,2,16,320,12928,985088,140861440,38451150848,20403322617856,
%T A171212 21307854867660800,44110759073910095872,181739941085108158595072,
%U A171212 1493546441998961207249207296,24512116566896662943648857456640
%N A171212 G.f.: A(x) satisfies A(x) = x + x*A(A(2*x)).
%C A171212 More generally, if F(x) = x + x*F(F(qx)), then
%C A171212 F(x) = x + x*F(qx) + x*F(qx)*F(qF(qx)) + x*F(qx)*F(qF(qx))*F(qF(qF(qx))) +...
%C A171212 with a simple solution at q=1/2:
%C A171212 F(x) = x/(1-x/2) satisfies: F(x) = x + x*F(F(x/2)).
%e A171212 G.f.: A(x) = x + 2*x^2 + 16*x^3 + 320*x^4 + 12928*x^5 +...
%e A171212 A(A(x)) = x + 4*x^2 + 40*x^3 + 808*x^4 + 30784*x^5 + 2200960*x^6 +...+ a(n)*x^n/2^(n-1) +...
%e A171212 As a formal series involving products of iterations of the g.f.,
%e A171212 A(x) = x + x*A(2x) + x*A(2x)*A(2A(2x)) + x*A(2x)*A(2A(2x))*A(2A(2A(2x))) +...
%e A171212 which, upon replacing x with A(2x), yields:
%e A171212 A(A(2x)) = A(2x) + A(2x)*A(2A(2x)) + A(2x)*A(2A(2x))*A(2A(2A(2x))) +...
%e A171212 thus A(x) = x + x*A(A(2x)).
%o A171212 (PARI) {a(n,q=2)=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 A171212 Cf. A171213 (q=3), A171214 (q=1/3).
%K A171212 nonn
%O A171212 0,3
%A A171212 _Paul D. Hanna_, Dec 08 2009
%E A171212 a(0) = 0 added by _Jason Yuen_, Feb 07 2025