Asymptotical behavior of this sequence limit

Question

A sequence is defined as $f(n)=\sum_{i=0}^nC_n^i\cdot i^{n-i}$, where $C_n^i$ is the combination number choosing $i$ from $n$. I would like to know how does the limit of this sequence behave like. According to numerical computation, it is potentially exponential.

Answer 1

According to $\text{A000248}$ there are asymptotics (Harris and Schoenfeld, 1968) given by

$$f(n) \sim \sqrt{\frac{r+1}{2\pi(n+1)(r^2+3r+1)}}\,\exp\left(\frac{n+1}{r+1}\right)\,\frac{n!}{r^n},$$

where $r$ is the root of $r(r+1)\exp(r) = n+1$.