firewall troubles, possibly lost email

[N. J. A. Sloane]

Paul wrote 
>       Sequences  A120588 - A120607  are examples of 
> solutions to functional equations of the form: 
>     r*A(x) = c + b*x + A(x)^n 

Some more examples of g.f.s satisfying a cubic are below.
Those that have a closed form all are products of binomials
so I guess the pattern continues, although I couldn't find
examples of higher than cubic g.f. solutions with closed form
in the OEIS.


ralf

A001764
A007863
A036759
A036765
A078531
A088927...

and a whole bunch by Emeric Deutsch:

%F A102403 G.f.=G=G(z) satisfies z^3*G^3+z(1-z)G^2-G+1=0.
%F A128729 G.f.=G=G(z) satisfies z^2*G^3-z(2-z)G^2+(1-z^2)G-1+z+z^2 =0.
%F A067955 a(n)=(1/n)sum(binomial(n, j)binomial((n-3-j)/2, j-1), j=1..floor((n-1)/3)) g.f. G(z) satisfies (1+z)G^3-zG^2-G+z=0
%F A120984 a(n)=(1/(n+1)*sum(3^(3j-n)*binomial(n+1,j)*binomial(j,n-2j), j=0..n+1). G.f.=G(z) satisfies G=1+3z^2*G^2+z^3*G^3.
%F A120985 a(n)=(1/(n+1))*sum(3^(n-3j)*binomial(n+1,2j+1)*binomial(n-2j,j), j=0..n/2). G.f.=G(z) satisfies G=1+3zG + z^3*G^3.
%F A128725 G.f.=G=G(z) satisfies z^2*G^3-2zG^2+(1+z-z^2)G-1=0.
%F A128736 G.f.=G=G(z) satisfies zG^3=(1-2z)(G-1)(2G-1).



ralf