Signed Stirling Number of the First Kind of Number with Self

Theorem

 * $\map s {n, n} = 1$

where $\map s {n, n}$ denotes a signed Stirling number of the first kind.

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


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

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

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

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 Number with Self
 * Stirling Number of the Second Kind of Number with Self


 * Particular Values of Signed Stirling Numbers of the First Kind