 Solution

At a nonpositive integer, the branch disappears and the collapsed Hankel contour extracts the residue at zero. For $s=0$,For $s=-2$, the coefficient of $t^{-1}$ inis $-1/4$; multiplication by $\Gamma (3)=2$ givesWhen $s=-2n-1$, the integrand is $t^{-2n-2}$ times an even power series, so it has no $t^{-1}$ coefficient. Hence