Binomial tail asymptotics

Let $$a_{n}=\sum_{k=2}^{n}\binom{n}{k}\frac{2^{k}(n-2)^{n-k}}{n^{n}},\quad n=2,3,….$$ Then, $e^{2}\lim_{n\to\infty}(1-a_{n})$ is equal to __________.