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:
 * $(1): \quad n! = a_0 + a_1 n + a_2 n \paren {n - 1} + a_3 n \paren {n - 1} \paren {n - 2} + \cdots$

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


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

Then: