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.
$\blacksquare$