How to find a closed form of this simple factorial sequence

Question

$$S_1=1\\ S_n=n!+S_{n-1}$$ Is there a simple way to express $S_n$ without summing up all the previous terms? Sorry I haven't put any effort in the problem but I don't know where to start.

So this means that $$S_1=1\\ S_2=3\\ S_3=9\\ S_4=33$$ and so on. Thanks in advance.

Answer 1

According to a CAS, there is effectively a closed form $$S_n=-(-1)^n \Gamma (n+2)~ \text{Subfactorial}[-n-2]-2 ~\text{Subfactorial}[-3]+1$$ in which function $\text{Subfactorial}[k]$ gives the number of permutations of $n$ objects that leave no object fixed.

I bet that this will not be very practical.

Answer 2

$S_n=1!+2!+\cdots + n!$. Now, a detailed discussion of this sum can be obtained here.

Answer 3

OEIS sequence A007489. No "closed form" is listed.