Residues of the Gamma function (${\mathrm{Res}}_{s=-j}\Gamma (s)=(-1{)}^j/j!$)

The Gamma function recurrence gives $\Gamma (s)=\Gamma (s+j+1)/(s(s+1)\cdots (s+j))$. Its numerator equals one at $s=-j$, so the residue there is $(-1{)}^j/j!$ for every nonnegative integer $j$.