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

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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:

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

where $\ds {n \brack n - 1}$ 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:

$\ds {n \brack n} = -\map s {n, n}$

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

$\blacksquare$


Also see