%I #7 Jun 29 2018 09:23:45
%S 1,2,10,106,2130,79442,5581914,754371386,199193444258,103781218984098,
%T 107308976428238250,220948088846408617994,907652841888542054277618,
%U 7447285848965361047618906866,122120561639979483596993367427066,4003478037366868501269046319075211994,262435985468992467742766325045697308601154,34402470626949826173994511431148770576433889602
%N G.f. A(x) satisfies: A(x) = x + x*A(x + A(x)).
%F G.f. A(x) satisfies:
%F (1) A(x) = x + x*A(x + A(x)).
%F (2) A(x) = Sum_{n>=0} Product_{k=0..n} B^k(x), where B(x) = x + A(x) and B^n(x) denotes the n-th iteration of B(x).
%e G.f.: A(x) = x + 2*x^2 + 10*x^3 + 106*x^4 + 2130*x^5 + 79442*x^6 + 5581914*x^7 + 754371386*x^8 + 199193444258*x^9 + 103781218984098*x^10 + ...
%e such that A(x) = x + x*A(x + A(x)).
%e RELATED SERIES.
%e Define B(x) = x + A(x), then
%e A(x) = x + x*B(x) + x*B(x)*B(B(x)) + x*B(x)*B(B(x))*B(B(B(x))) + x*B(x)*B(B(x))*B(B(B(x)))*B(B(B(B(x)))) + ...
%e where
%e B(x) = 2*(x + x^2 + 5*x^3 + 53*x^4 + 1065*x^5 + 39721*x^6 + ...);
%e B(B(x)) = 4*(x + 3*x^2 + 29*x^3 + 559*x^4 + 20393*x^5 + 1415339*x^6 + ...);
%e B(B(B(x))) = 8*(x + 7*x^2 + 133*x^3 + 4939*x^4 + 348025*x^5 + ...);
%e B(B(B(B(x)))) = 16*(x + 15*x^2 + 565*x^3 + 41315*x^4 + 5738713*x^5 + ...);
%e B(B(B(B(B(x))))) = 32*(x + 31*x^2 + 2325*x^3 + 337683*x^4 + 93186713*x^5 + ...);
%e etc.
%o (PARI) {a(n) = my(A=x); for(i=1,n, A = x + x*subst(A,x,x + A + x*O(x^n))); polcoeff(A,n)}
%o for(n=1,20,print1(a(n),", "))
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 28 2018