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A182224 G.f. satisfies: A(x) = 1 + x*A(x) * A(A(x) - 1).

%I A182224 #9 Apr 20 2012 09:59:46
%S A182224 1,1,2,7,35,220,1622,13480,123212,1218694,12898292,144890911,
%T A182224 1717072304,21367220392,278174617499,3777454890226,53372573914742,
%U A182224 783004637781214,11905653060557285,187325244287570431,3045651218248945454,51103551998888439679,883901254149820933025
%N A182224 G.f. satisfies: A(x) = 1 + x*A(x) * A(A(x) - 1).
%F A182224 Given g.f. A(x), define G(x) by G(x)/x = g.f. of A088717, then G(x) satisfies:
%F A182224 (1) G(x) = x + x*G(G(x)^2/x),
%F A182224 (2) G(x) = Series_Reversion(x/A(x)),
%F A182224 (3) G(x) = x/Series_Reversion(x*A(x)) - 1,
%F A182224 (4) G(x) = x*A(G(x)),
%F A182224 (5) G(x) = A(x/(1+G(x))) - 1,
%F A182224 (6) A(x) = 1 + G(x*A(x)).
%e A182224 G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 35*x^4 + 220*x^5 + 1622*x^6 +...
%e A182224 such that A(x) = 1 + x*A(x)*A(A(x)-1) where:
%e A182224 A(A(x)-1) = 1 + x + 4*x^2 + 22*x^3 + 148*x^4 + 1147*x^5 + 9901*x^6 +...
%e A182224 Let G(x) satisfy G(x/A(x)) = x, then G(x)/x = g.f. of A088717, where
%e A182224 G(x) = x + x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 596*x^6 + 4785*x^7 +...
%e A182224 A(G(x)) = 1 + x + 3*x^2 + 14*x^3 + 84*x^4 + 596*x^5 + 4785*x^6 +...
%e A182224 A(x/(1+G(x))) = 1 + x + x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 596*x^6 + 4785*x^7 +...
%e A182224 G(x*A(x)) = x + 2*x^2 + 7*x^3 + 35*x^4 + 220*x^5 + 1622*x^6 +...
%o A182224 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*A*subst(A,x,A-1)+x*O(x^n));polcoeff(A,n)}
%o A182224 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+subst(serreverse(x/A),x,x*A)+x*O(x^n));polcoeff(A,n)}
%o A182224 for(n=0,25,print1(a(n),", "))
%Y A182224 Cf. A088717.
%K A182224 nonn
%O A182224 0,3
%A A182224 _Paul D. Hanna_, Apr 19 2012