$$\cases{T(0)=3\\ 3T(n)=nT(n-1)+2n!}
$$
|n|0|1|2|
|-|-|-|-|
|T(n)|$3$|$\frac{5}{3}$|$\frac{22}{9}$

$$\begin{gather}

a_n=3\\
b_n=n\\
c_n=2n!\\
s_{n}=\cfrac{3^{n-1}}{n!}\\
T_{n}=\cfrac{1}{\frac{3^{n-1}}{n!}\cdot 3} \cdot\left(1\cdot1\cdot3+\sum\limits_{k=1}^{n}\frac{3^{k-1}}{k!}\cdot 2k!\right)=\frac{n!}{3^{n}}\left(3+\sum\limits_{k=1}^{n}3^{k-1}\cdot \frac{2}{3}\right)=\\=\frac{n!}{3^{n}}\left(3+ \frac{2}{3}\cdot \frac{3-3^{n-1}}{1-3}\right)=\frac{n!}{3^{n}}\left(2+3^{n}\right)
\end{gather}$$