By multinomial expansion formula, we know that $$ \sum_{p_1 + \cdots + p_k = r} \binom{r}{p_1,\ldots,p_k} = k^r, $$ where the multinomial coefficient is defined by $ \binom{r}{p_1, \ldots, p_k} := \frac{r!}{p_1!\cdots p_k!}$. Here is my question:

How can we find the sum $$ \sum_{p_1 + \cdots + p_k=r} \binom{r}{p_1,\ldots,p_k} $$ with the restriction that all $ p_j $'s are even? This sum shows up in some multiple commutators of Hilbert space operators. Any hint is greatly appreciated.