Suppose there are $n$ distinct balls in a bag and they are drawn with replacement until the first repeat. Let $X$ be the number of balls drawn. I have shown that the distribution is unimodal and that the mode is asymptotically equal to $\sqrt n$. The expectation of the distribution is derived here to be
$$ \sum_{k \ge 0} \binom{n}{k} \frac{k!}{n^k} $$
I believe that this also should asymptotically equal $\sqrt n$ or $C \sqrt n$ for some $C$, and my numerical experiments have confirmed this. Is this true and why?
A derivation can be found in this paper.
https://www.sciencedirect.com/science/article/pii/0377042793E0258N?via%3Dihub