0,3

G.f.: A(x) = f(x*A(x)) = (1-1/f(x))/x where f(x) is the g.f. of A088713.

Given g.f. A(x), then B(x)=x*A(x) satisfies 0=f(x, B(x), B(B(x))) where f(a0, a1, a2)=a0-a1+a1*a2 . - Michael Somos May 21 2005

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

G.f. satisfies: A(x) = (1/x)*Series_Reversion(x - x^2*A(x)).

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2009: (Start)

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then

a(n,m) = Sum_{k=0..n} m*C(n+k+m,k)/(n+k+m) * a(n-k,k).

(End)

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 + xBC;

C = B + xCD;

D = C + xDE;

E = D + xEF ; ...

(PARI) {a(n)=local(A); if(n<0, 0, n++; A=x+O(x^2); for(i=2, n, A=x/(1-subst(A, x, A))); polcoeff(A, n))} /* Michael Somos May 21 2005 */

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

(PARI) {a(n)=local(A); if(n<0, 0, A=1+x+O(x^2); for(i=0, n, A=(1/x)*serreverse(x-x^2*A)); polcoeff(A, n))}

(PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(n+k+m, k)/(n+k+m)*a(n-k, k))))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2009]

Cf. A088713.

Apart from signs, same as A067145. - Philippe DELEHAM, Jun 18 2006

Cf. A002449, A030266, A087949, A088717, A091713, A120971.

Sequence in context: A074534 A153395 A067145 this_sequence A088368 A007808 A104989

Adjacent sequences: A088711 A088712 A088713 this_sequence A088715 A088716 A088717

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 12 2003, May 22 2008