Relation between Signed and Unsigned Stirling Numbers of the First Kind

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Theorem

Let $m, n \in \Z_{\ge 0}$ be positive integers.

Then:

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

where:

$\displaystyle \left[{n \atop m}\right]$ denotes an unsigned Stirling number of the first kind
$s \left({n, m}\right)$ denotes a signed Stirling number of the first kind.


Proof