# Is there any upper bound of this sum?

$a_1,a_2,\ldots,a_n,k$ are all integers. Is there any upper bound of the following sum

$$\sum_{a_1+a_2+\cdots+a_n=k\textrm{ and } a_1,a_2,\ldots,a_n\ge 0} \frac{1}{a_1!a_2!\cdots a_n!},$$ which is a function of $n$ and $k$?

Recall multinomial theorem: $$(x_1+x_2+ \cdots+x_n)^k = \sum_{\overset{a_1+\cdots+a_n=k}{a_i \in \mathbb{N}}} \dfrac{k!}{a_1! a_2! \cdots a_n!}x_1^{a_1}x_2^{a_2} \cdots x_n^{a_n}$$ Taking $x_1=x_2=\cdots=x_n=1$, we obtain $$\sum_{\overset{a_1+\cdots+a_n=k}{a_i \in \mathbb{N}}} \dfrac1{a_1! a_2! \cdots a_n!} = \dfrac{n^k}{k!}$$