The following summation came up while I was exploring some multivariate generalizations of the asymmetric Laplace distribution sometimes used in Bayesian quantile regression. $n$ is a positive integer, and $\lambda$ is a positive real.
$$ \sum_{k=0}^{n-1} \binom{n}{k} \frac{(n - k)^{n - k}}{\lambda^k (n + \lambda k)^{n - k}} $$
Does this reduce to a closed form? It looks like some kind of Binomial theorem style identity could apply, but I haven't been able to figure anything out. I've tried looking at $\lambda=1$ as a special case in the hopes it would be simpler, but no luck.