Unsigned Stirling Number of the First Kind of n with n-2

Theorem
Let $n \in \Z_{> 0}$ be an integer greater than $0$.

Then:
 * $\displaystyle \left[{n \atop n - 2}\right] = \binom n 4 + 2 \binom {n + 1} 4$

where:
 * $\displaystyle \left[{n \atop n - 2}\right]$ denotes an unsigned Stirling number of the first kind
 * $\dbinom n 4$ denotes a binomial coefficient.

Proof
The proof proceeds by induction.

Basis for the Induction
For all $n \in \Z_{\ge 2}$, let $P \left({n}\right)$ be the proposition:
 * $\displaystyle \left[{n \atop n - 2}\right] = \binom n 4 + 2 \binom {n + 1} 4$

$P \left({2}\right)$ 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:
 * $\displaystyle \left[{k \atop k - 2}\right] = \binom k 4 + 2 \binom {k + 1} 4$

from which it is to be shown that:
 * $\displaystyle \left[{ {k + 1} \atop k - 1}\right] = \binom {k + 1} 4 + 2 \binom {k + 2} 4$

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:
 * $\displaystyle \forall n \in \Z_{\ge 1}: \left[{n \atop n - 1}\right] = \binom n 4 + 2 \binom {n + 1} 4$

Also see

 * Stirling Number of the Second Kind of n with n-2


 * Particular Values of Unsigned Stirling Numbers of the First Kind