We know that the OGF $T(z)$ of convex polygonials satisfies the relation $$T(z) = G(z,T(z)) := 1 + zT(z)^2.$$ Why does $G(z,w)$ not satisfy the requirements of the implicits function scheme? Find a function $H(z,T(z))$ so that the implicit function scheme is applicable.
Remark: You can find the relevant section from the Flajolet & Sedgewick book page 467, 468 below.
I know that $$T(z) = \frac{1-\sqrt{1-4z}}{2z}$$
and that $T(z)$ is analytic in $\lvert z \rvert < 1/4$. However, I am confused on how to check the conditions of the implicit function scheme on $G(z,T(z))$, because I do not see how to obtain the coefficients $g_{m,n}$. Could you please explain this to me?
