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

Proof
Let $n$ be a non-negative integer.

We assume a solution of the form:
 * $(1): \quad n! = a_0 + a_1 n + a_2 n \left({n - 1}\right) + a_3 n \left({n - 1}\right) \left({n - 2}\right) + \cdots$

We can express $(1)$ using binomial coefficients:


 * $(2): \quad n! = \displaystyle \sum_k \dbinom n k k! a_k$

Then: