Unsigned Stirling Number of the First Kind of n+1 with 1

Theorem
Let $n \in \Z_{\ge 0}$.

Then:
 * $\displaystyle \left[{n + 1 \atop 1}\right] = n!$

where:
 * $\displaystyle \left[{n + 1 \atop 1}\right]$ denotes an unsigned Stirling number of the first kind
 * $n!$ denotes $n$ factorial.

Proof
The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $P \left({n}\right)$ be the proposition:
 * $\displaystyle \left[{n + 1 \atop 1}\right] = n!$

$P \left({0}\right)$ is the case:

Basis for the Induction
$P \left({1}\right)$ is the case:

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $P \left({k}\right)$ is true, where $k \ge 2$, then it logically follows that $P \left({k + 1}\right)$ is true.

So this is the induction hypothesis:
 * $\displaystyle \left[{k + 1 \atop 1}\right] = k!$

from which it is to be shown that:
 * $\displaystyle \left[{k + 2 \atop 1}\right] = \left({k + 1}\right)!$

Induction Step
This is the induction step:

So $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \forall n \in \Z_{\ge 0}: \left[{n + 1 \atop 1}\right] = n!$

Also see

 * Signed Stirling Number of the First Kind of n+1 with 1
 * Stirling Number of the Second Kind of n+1 with 1


 * Particular Values of Unsigned Stirling Numbers of the First Kind