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%I #6 Aug 03 2012 14:53:30
%S 1,3,9,81,891,11907,184437,3199581,60932007,1257133527,27836230041,
%T 656867748537,16429561047891,433686821472747,12038953175046909,
%U 350402975398982133,10665927632978564895,338769129913521564735,11205026468737167058785
%N G.f. satisfies: A(x) = x + 2*x^2 + x*A(A(A(x))).
%C The (1/3)-iteration of the g.f. equals an integer series (A215115).
%F G.f. satisfies: A(x) = G(x)/G(G(x)) - 1 - G(G(x)) where A(G(x)) = x.
%e G.f.: A(x) = x + 3*x^2 + 9*x^3 + 81*x^4 + 891*x^5 + 11907*x^6 + 184437*x^7 +...
%e where
%e A(A(A(x))) = x + 9*x^2 + 81*x^3 + 891*x^4 + 11907*x^5 + 184437*x^6 +...
%e Related expansions.
%e Let C(C(C(x))) = A(x), then C(x) is an integer series where:
%e C(x) = x + x^2 + x^3 + 19*x^4 + 163*x^5 + 2269*x^6 + 34093*x^7 +...
%e where the coefficients of C(x) are congruent to 1 modulo 9.
%o (PARI) {a(n)=local(A=x+3*x^2); for(i=1, n, A=x+2*x^2+x*subst(A, x, subst(A, x, A+x*O(x^n)))); polcoeff(A, n)}
%o for(n=1, 31, print1(a(n), ", "))
%Y Cf. A215115, A213010, A215116, A215118.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 03 2012
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