1,3

G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+2)) for n>0 with F(x,0)=1. - Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2007

G.f.: A(x) = G(x)/[1 + G(G(x))] where G(x) = A(A(x)) = g.f. of A141117.

G.f.: A(x) = Series_Reversion[ x/(1 + A(A(x))) ].

Comment from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2007: G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:

A = 1 + xC;

B = A*(1 + xD);

C = B*(1 + xE);

D = C*(1 + xF);

E = D*(1 + xG) ; ...

The solution to the variables in the system of equations are

A=A(x), B=A(A(x)), C=A(A(A(x))), D=A(A(A(A(x)))), etc., where:

A(x) = x + x^2 + 3*x^3 + 15*x^4 + 99*x^5 + 781*x^6 + 7001*x^7 +...

A(A(x)) = x + 2*x^2 + 8*x^3 + 46*x^4 + 330*x^5 + 2756*x^6 +...

A(A(A(x))) = x + 3*x^2 + 15*x^3 + 99*x^4 + 781*x^5 + 7001*x^6 +...

A(A(A(A(x)))) = x + 4*x^2 + 24*x^3 + 180*x^4 + 1564*x^5 +...

A(A(A(A(A(x))))) = x + 5*x^2 + 35*x^3 + 295*x^4 + 2815*x^5 +...

(PARI) {a(n)=local(A); A=x+x^2; for(i=3, n, A=x+x*subst(A, x, subst(A, x, A))+x*O(x^n)); polcoeff(A, n, x)}

Cf. A002449, A030266, A087949, A088714, A088717, A120971.

Cf. A141117.

Sequence in context: A128081 A046635 A140286 this_sequence A156106 A111546 A152402

Adjacent sequences: A091710 A091711 A091712 this_sequence A091714 A091715 A091716

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 31 2004, Jun 04 2008