Signed Stirling Number of the First Kind of n with n-1

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

Then:
 * $s \left({n, n - 1}\right) = -\dbinom n 2$

where:
 * $s \left({n, n}\right)$ denotes a signed Stirling number of the first kind
 * $\dbinom n 2$ denotes a binomial coefficient.

Proof
From Relation between Signed and Unsigned Stirling Numbers of the First Kind:


 * $\displaystyle \left[{n \atop n - 1}\right] = \left({-1}\right)^{n + n - 1} s \left({n, n - 1}\right)$

where $\displaystyle \left[{n \atop n - 1}\right]$ denotes an unsigned Stirling number of the first kind.

We have that:
 * $\left({-1}\right)^{n + n - 1} = \left({-1}\right)^{2 n - 1} = -1$

and so:
 * $\displaystyle \left[{n \atop n}\right] = -s \left({n, n}\right)$

The result follows from Unsigned Stirling Number of the First Kind of Number with Self.

Also see

 * Unsigned Stirling Number of the First Kind of n with n-1
 * Stirling Number of the Second Kind of n with n-1


 * Particular Values of Signed Stirling Numbers of the First Kind