I am looking at a special case of the binomial PDF, where
$$p = \frac{1}{n}$$
i.e.,
$$f(n,k) = \frac{n!}{k!(n-k)!} \left(\frac{1}{n}\right)^k \left(1-\frac{1}{n}\right)^{n-k}$$

I have observed empirically that in the limit that $n\rightarrow \infty$, the two-variable PDF appears to become a new one-variable PDF: $f(n,k)\rightarrow g(k)$.

I would like a closed form expression for this new function $g(k)$, if such an expression exists.