$\lim_{n \to \infty}\left(\sum_{k=0}^{n}\left(\frac{\left(k-n\right)^k}{k!}\cdot e^{n-k}\right)-2n\right)$

Evaluate $\lim\limits_{n \to \infty}\left(\sum\limits_{k=0}^{n}\left(\frac{\left(k-n\right)^k}{k!}\cdot e^{n-k}\right)-2n\right)$.

By plugging in large values for $n$ I noticed that the limit is most likely $\frac{2}{3}$, but I can't prove it.

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