Tilde notation of $\sum_{i=0}^n \frac{n!}{(n-i)!}$

Question

I'm trying to find the growth rate (tilde notation) of this function:

$$f(n) = \sum_{i=0}^n \frac{n!}{(n-i)!}$$

The $\sim$ notation is similar to the big O notation. However the $\sim$ notation is also interested in the coefficient of the fastest growing term.

For example: $3x^3 + 7x^2 - 90 \sim 3x^3$

Answer 1

Let $f(n)$ be the sum. You can show that $f(n)=\lfloor{en!}\rfloor$. See this MSE post for example

Answer 2

The growth is same as $n!$ which is $\approx\sqrt{2\pi n}(n/e)^{n}$ from Stirling approximation. The reason is $\sum_{i=0}^n \frac{1}{i!}$ is $O(1)$. In particular it is less than 3.