  The Poisson distribution of order $k$ is a special case of a compound Poisson
distribution. For $k=1$ it is the standard Poisson distribution. Our focus in
this note is for $k\ge2$. For sufficiently small values of the rate parameter
$\lambda$, both the median and mode equal zero. The median is zero if and only
if $\lambda \le (\ln2)/k$. The supremum value of $\lambda$ for the mode to be
zero is known only for small values of $k$. This note presents results for the
"first double mode" by which is meant the first occasion (smallest value of
$\lambda$) the distribution is bimodal, with modes at $0$ and $m>0$. Next, an
almost complete answer is supplied to the question "which positive integers
cannot be modes of the Poisson distribution of order $k$?" The term "almost
complete" signifies that some parts of the answer are conjectures based on
numerical searches. However, if the conjectures are proved to be correct, the
solution presented in this note is complete: all parameter values are covered.
