I would appreciate any help on the following two problem:
Given the symbolic equation $\mathcal{T}=\mathcal{Q}^{[\ast]}$ and $\mathcal{Q}=Δ∗\mathcal{T}∗\mathcal{T}$, I am trying to build the exponential generating functions corresponding to these equations (as described, for example, in the book "Analytic Combinatorics" http://algo.inria.fr/flajolet/Publications/book.pdf) and read $T_n$ as the coefficient $n![z^n]$ of the generating function.
As far as I've understood it, the equations should correspond with (T(z), Q(z) denoting the exponential generating function):
$Q(z)=z⋅T(z)⋅T(z)$ and $T(z)=exp(Q(z))$
Now I'm trying to apply the Lagrange–Bürmann formula. So I've expressed z as $z=\frac{log(T)}{T^2}$=:$\frac{x}{Φ(x)}$ and do $[z^n]T(z)$=$[z^n]f(log(T))$=$\frac{1}{n}[x^{n−1}]f′(x)(\Phi(x))^n$ = ... = $\frac{1}{n}[x^n](\sum_{k\geq 0}\frac{x^k}{k!})^{2n−2}$
Since I see no way, to read the coefficient now, I guess I've probably been mistaken somewhere. Does anyone find my mistake?