For $n \ge 0$, let $h_n$ be the number of ways of taking $n$ (distinguishable) rabbits, putting them into identical cages with one to three rabbits per cage and then ordering the cages in a row. Find an exponential generating function for $h_n$ (assuming $h_0 = 1$).
I know if the ordering of the cages does not matter, then generating function would be $h_n = h_{n-1}+(n-1)h_{n-2}+(n-2)(n-1)h_{n-3}$. How can I include the ordering of the cages in the function?
I computed the first nine values of $h_n$ $(n\geq1)$ recursively and obtained the sequence $$ 1, 3, 13, 74, 530, 4550, 45570, 521640, 6717480\ .$$ This is sequence A189886 in OEIS. There is a reference to sequences of sets, no set containing more than three elements. But I didn't study the details.