%I #4 Dec 14 2012 16:17:02
%S 1,1,2,7,26,104,436,1894,8444,38418,177656,832548,3945156,18871524,
%T 91003360,441927367,2159282462,10607708284,52363342484,259601860898,
%U 1292041756732,6453179670344,32334136480656,162487089008766
%N G.f. satisfies: A(x) = x + A(A(A(x)^2)).
%F G.f. A(x) satisfies:
%F (1) A( x - A(A(x^2)) ) = x.
%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(A(x^2))^n / n!.
%F (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(A(x^2))^n/x / n! ).
%e G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 104*x^6 + 436*x^7 +...
%e The series reversion of A(x) = x - A(A(x^2)), where
%e A(A(x^2)) = x^2 + 2*x^4 + 6*x^6 + 25*x^8 + 116*x^10 + 574*x^12 + 2972*x^14 +...
%e The g.f. satisfies the series:
%e A(x) = x + A(A(x^2)) + d/dx A(A(x^2))^2/2! + d^2/dx^2 A(A(x^2))^3/3! + d^3/dx^3 A(A(x^2))^4/4! +...
%e as well as the logarithmic series:
%e log(A(x)/x) = A(A(x^2))/x + [d/dx A(A(x^2))^2/x]/2! + [d^2/dx^2 A(A(x^2))^3/x]/3! + [d^3/dx^3 A(A(x^2))^4/x]/4! +...
%o (PARI) {a(n)=local(A=x+x^2);for(i=1,n,A=x+subst(A,x,subst(A,x,A^2+x*O(x^n))));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - subst(A,x,subst(A,x,x^2+x*O(x^n))) )); polcoeff(A, n))}
%o for(n=1,25,print1(a(n),", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, subst(A,x,subst(A,x,x^2+x*O(x^n)))^m)/m!)+x*O(x^n)); polcoeff(A, n)}
%o for(n=1,25,print1(a(n),", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, subst(A,x,subst(A,x,x^2+x*O(x^n)))^m/x)/m!)+x*O(x^n))); polcoeff(A, n)}
%o for(n=1,25,print1(a(n),", "))
%Y Cf. A141371, A141372.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jun 28 2008
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