Values of some "natural" sums over multiindices with a given absolut value

Question

I'd like to know if there is a nice closed expression in terms of $j$ and $k$ of the sum $$ S_{j,k}:=\sum_{(i_1,\ldots,i_k)\in \mathbb{N}^k_0:\\i_1+\cdots+i_k=j}\frac{1}{i_1!\cdots i_k!}. $$ Furthermore, given $m\in \{0,\ldots,k\}$, I am interested in a closed expression for the values $$ S^m_{j,k}:=\sum_{(i_1,\ldots,i_k)\in \mathbb{N}^k_0:\\i_1+\cdots+i_k=j,\\ m\text{ of the numbers } \\i_1,\ldots,i_k \text{ are odd} }\frac{1}{i_1!\cdots i_k!}, $$ but I guess that this is hopeless. In both cases, I am happy to assume that $j$ and $k$ are even.

Answer 1

$e^x = \sum\limits_{n = 0}^{\infty} \frac{x^n}{n!}$, but then: $e^x\cdot e^x...\cdot e^x = e^{kx} = \sum\limits_{n = 0}^{\infty} \frac{k^n\cdot x^n}{n!}$ and we see that $S_{j, k} = \frac{k^j}{j!}$, because left hand side is a product of $k$ series and the coefficient before $x^j$ is what we need.

edit: For the second one: see comments by @MikeErnest below.