有标号有根树计数

初学拉格朗日反演/kel

首先先写一下 EGF：

设其 EGF 为 $T(x)$，则有 $T(x)=xe^{T(x)}$．

$$\begin{aligned} T(x)=xe^{T(x)} \\ G(x)=T^{-1}(x)=\frac{x}{e^x} \\ H(x)=e^x \\ n![x^n]T(x)=&\ n![x^{n-1}]e^{T(x)} \\=&\ n![x^{n-1}]H(T) \end{aligned}$$

拉格朗日反演

$$\begin{aligned} &n![x^{n-1}]H(T) \\ =&n!\frac{1}{n-1}[x^{n-2}]H'(x)(\frac{x}{G})^{n-1} \\ =&n!\frac{1}{n-1}[x^{n-2}]e^x(e^x)^{n-1} \\ =&n!\frac{1}{n-1}[x^{n-2}]e^{nx} \\ =&n!\frac{1}{n-1}n^{n-2}\frac{1}{(n-2)!} \\ =&n^{n-1} \end{aligned}$$