Factorial moments of the Poisson distribution of order $k$

The factorial moments of the standard Poisson distribution are well known. The present note presents an explicit combinatorial sum for the factorial moments of the Poisson distribution of order $k$. Unlike the standard Poisson distribution (the case $k=1$), for $k>1$ the structure of the factorial moments is much more complicated. Some properties of the factorial moments of the Poisson distribution of order $k$ are elucidated in this note.