Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial/Proof

Theorem

 * $n! = a_0 + a_i n + a_2 n \left({n - 1}\right) + a_2 n \left({n - 1}\right) \left({n - 2}\right) + \cdots$

where:
 * $\displaystyle a_m = \sum_{k \mathop = 0}^m \dfrac {\left({-1}\right)^k} {k!}$

Proof
Let:
 * $\displaystyle (1): \quad n! = \sum_k \binom n k k! \, a_k$

Then: