0,3

Coefficient of x^n of A(x)^2 equals coefficient of x^n in (1+x*A(x))^(n+1)/(n+1).

Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy: A = 1 + xA^2*B, B = 1 + x(AB)^2*C, C = 1 + x(ABC)^2*D, D = 1 + x(ABCD)^2*E, etc., then B(x)=A(x*A(x)^2), C(x)=B(x*A(x)^2), D(x)=C(x*A(x)^2), etc., where A(x) = 1 + x*A(x)^2*A(x*A(x)^2) is the g.f. of this sequence (see table A128330).

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

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 + xAB;

B = A*(1 + xBC);

C = B*(1 + xCD);

D = C*(1 + xDE);

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

(PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+x*A^2*subst(A, x, x*A^2+x*O(x^n))); polcoeff(A, n)}

Cf. A128330, A030266.

Cf. A002449, A030266, A087949, A088714, A091713, A120971.

Sequence in context: A154757 A074535 A005700 this_sequence A111538 A088716 A005189

Adjacent sequences: A088714 A088715 A088716 this_sequence A088718 A088719 A088720

nonn,eigen

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 12 2003 and Mar 10 2007