This is a boring, technical question that I stumbled upon while making a contribution to Sage. I would still like to hear a constructive answer so hopefully the question does not get closed. The ...
Sequence A003238 of the OEIS counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins 1, 1, 2, 3, 5, 6, 10, 11, 16, ... and it is ...