Moments of the Poisson distribution of order $k$

The factorial moments of the standard Poisson distribution are well known and are simple, but the raw moments are considered to be more complicated (Touchard polynomials). The present note presents a recurrence relation and an explicit combinatorial sum for the raw moments of the Poisson distribution of order $k$. Unlike the standard Poisson distribution (the case $k=1$), for $k>1$ the structure of the raw and factorial moments have many similarities and the raw moments are not more complicated (formally, at least) than the factorial moments. We remark briefly on the central moments (i.e.~moments centered on the mean) of the Poisson distribution of order $k$.