Is there an explicit formula for the following quantity?

$$f_m(a_1,\ldots,a_n):=\sum_{\substack{k_1+\ldots+k_n=m \\ k_1,\ldots,k_n\in \mathbb{N}}} k_1^{a_1}\ldots k_n^{a_n}\ ,\hspace{1cm} m,a_1,\ldots,a_n\in \mathbb{N}$$

(for instance $f_m(0,\ldots,0)$ is simply the number of compositions of m into n parts, $f_m(1,1)=\frac{m(m+1)(m-1)}{6}$ and so on). I would like an answer both for the case where the $k_i$'s can and cannot attain the value $0$, if possible.