  We prove that the rational Euler characteristic of $\operatorname{Out}(F_n)$
is always negative and its asymptotic growth rate is $\Gamma(n-
\frac32)/\sqrt{2\pi} \log^2 n$. This settles a 1987 conjecture of J. Smillie
and the second author. We establish connections with the Lambert $W$-function
and the zeta function.
