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

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Theorem

\(\ds n!\) \(=\) \(\ds \sum_{k \mathop \ge 0} \dfrac { {!k} \, n^{\underline k} } {k!}\)
\(\ds \) \(=\) \(\ds \dfrac { !0 \times n^{\underline 0} } {0!} + \dfrac { {!1} \times n^{\underline 1} } {1!} + \dfrac { {!2} \times n^{\underline 2} } {2!} + \dfrac { {!3} \times n^{\underline 3} } {3!} + \cdots\)
\(\ds \) \(=\) \(\ds 1 + \left({1 - \dfrac 1 {1 !} }\right) n + \left({1 - \dfrac 1 {1 !} + \dfrac 1 {2 !} }\right) n \left({n - 1}\right) + \left({1 - \dfrac 1 {1 !} + \dfrac 1 {2 !} - \dfrac 1 {3 !} }\right) n \left({n - 1}\right) \left({n - 2}\right) + \cdots\)


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:

\(\ds \sum_n n! \binom m n \paren {-1}^{n - k}\) \(=\) \(\ds \sum_n \paren {\sum_k \binom n k k! \, a_k} \binom m n \paren {-1}^{n - k}\) substituting for $n!$ from $(2)$
\(\ds \) \(=\) \(\ds \sum_k k! \, a_k \sum_n \binom n k \binom m n \paren {-1}^{m - n}\) changing order of summation
\(\ds \) \(=\) \(\ds \sum_k k! \, a_k \delta_{k m}\) Corollary to Sum over $k$ of $\dbinom r k \dbinom {s + k} n \paren {-1}^{r - k}$
\(\ds \) \(=\) \(\ds m! \, a_m\) as the $m$th term is the only one left standing
\(\ds \leadsto \ \ \) \(\ds a_m\) \(=\) \(\ds \sum_{n \mathop \ge 0} \paren {-1}^{m - n} \dfrac {n!} {m!} \binom m n\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^m \paren {-1}^{m - n} \dfrac {n!} {m!} \dfrac {m!} {n! \, \paren {m - n}!}\) Definition of Binomial Coefficient
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^m \frac {\paren {-1}^{m - n} } {\paren {m - n}!}\) simplifying
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^m \frac {\paren {-1}^n} {n!}\) Permutation of Indices of Summation

$\blacksquare$


Sources