Solve recurrence equation: $a_{n}=(n-1)(a_{n-1}+a_{n-2})$

Question

Find and solve recurrence equation $a_{n}$, where $a_{n}$ is the number of derangement in permutation $\pi\in S_{n}$, and $\forall_{i\in \mathbb{N}_{n}}\pi(i)\ne i$.
I managed to find the equation: $$ a_{n} = \begin{cases} 1, & \text{if $n$ = 0} \\ 0, & \text{if $n$ = 1} \\ (n-1)(a_{n-1}+a_{n-2}), & \text{if $n\ge2$} \end{cases}, $$ but I don't know how to solve it. Only managed to get to this: $A(x) = 1+\sum^\infty_{n=0}{(n+1)a_{n+1}x^{n+2}}+\sum^\infty_{n=0}{(n+1)a_{n}x^{n+2}}$.

Answer 1

the solution is given by the Gamma-function, $$a(n)=\frac{\Gamma (n+1,-1)-\Gamma (2,-1) \Gamma (n+1)}{e-\Gamma (2,-1)}$$