这是一份umass麻省大学 MATH 513作业代写的成功案例
$$
T_{n}=n U_{n} \quad \text { implying } \quad U_{n}=n^{n-2} \text {. }
$$
At generating function level, this combinatorial equality translates into
$$
U(z)=\int_{0}^{z} T(w) \frac{d w}{w},
$$
which integrates to give (take $T$ as the independent variable)
$$
U(z)=T(z)-\frac{1}{2} T(z)^{2} .
$$
Since $U(z)$ is the EGF of acyclic connected graphs, the quantity
$$
A(z)=e^{U(z)}=e^{T(z)-T(z)^{2} / 2},
$$
MATH 513 COURSE NOTES ：
Substitution (composition). The composition or substitution can be defined so that it corresponds a priori to composition of generating functions. It is formally defined as
$$
\mathcal{B} \circ \mathcal{C}=\sum_{k=0}^{\infty} \mathcal{B}{k} \times \operatorname{SET}{k}{\mathcal{C}},
$$
so that its EGF is
$$
\sum_{k=0}^{\infty} B_{k} \frac{(C(z))^{k}}{k !}=B(C(z))
$$