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