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:
 * $\map s {n, n - 1} = -\dbinom n 2$

where:
 * $\map s {n, k}$ 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:


 * $\displaystyle {n \brack n - 1} = \paren {-1}^{n + n - 1} \map s {n, n - 1}$

where $\displaystyle {n \brack n - 1}$ denotes an unsigned Stirling number of the first kind.

We have that:
 * $\paren {-1}^{n + n - 1} = \paren {-1}^{2 n - 1} = -1$

and so:
 * $\displaystyle {n \brack n} = -\map s {n, n}$

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