Proving a Recursion Using Induction I am trying to prove the following recursion.
$$a(n) = \left\{\begin{matrix}
n(a(n-1)+1) & \text{if } n \geq 1\\ 
0 & \text{if } n = 0
\end{matrix}\right.$$
is the series definition of $a(n)$. using this, I need to prove that
$$ a(n) = n!\bigg(\frac{1}{0!} + \frac{1}{1!} + \cdots + \frac{1}{(n-1)!}\bigg)$$
for $n \geq 1$ by induction on $n$.
I've found that the $n$ equals, for the first 5 terms, $2,5,16,65,326$. I think now I need to find a formula that describes these terms, and therefore $a(n)$. The problem is, I don't know where to start. Can anyone give me a hand?

For $m\ge1,$
If $a(m)=m!\left(\sum_{r=0^{m-}}\frac1{r!}\right)$
\begin{align}
a(m+1) & =(m+1)[a(m)+1] \\[6pt]
& =(m+1)[m!\left(\sum_{r=0^{m-}}\frac1{r!}\right)+1] \\[6pt]
& =(m+1)!\left(\sum_{r=0^{m-}}\frac1{r!}\right)+m+1 \\[6pt]
& =(m+1)!\left(\sum_{r=0^{m-}}\frac1{r!}\right)+\frac{(m+1)!}{m!}
\end{align}
