TITLE: Is there a natural relationship between OEIS A127670 and Cayley's tree formula?
QUESTION [7 upvotes]: I apologize in advance that this question must sound highly amateurish, but I am wondering if there is any connection between the formula https://oeis.org/A127670 , which counts the number of fixed $n$-cell polycubes that are proper in $n-1$ dimensions, and Cayley's tree formula.  The expression for the former is $$a_n = 2^n(n+1)^{n-2}$$ and the latter is $$b_n = n^{n-2}$$, which means that $$a_n = \frac{2^n}{n+1} b_{n+1}$$.  Is there any significance to this at all?  The reason I ask is that I'm working on a Cayley-type tree enumeration problem that yields the sequence $a_n$ according to numerical computations, and I am struggling to see the connection.

REPLY [14 votes]: Yes there is a connection. While $n^{n-2}$ counts the number of vertex labeled trees on $n$ vertices, the expression $2^n(n+1)^{n-2}$ counts the number of edge labeled trees on $n$ edges. There is a bijection between edge labeled trees on $n$ vertices and proper $(n-1)$-dimensional polycubes of size $n$. See lemma 2 (which combinatorially proves the relation $a_n=\frac{2^n}{n+1}b_{n+1}$) and theorem 1 in Formulae and growth rates of high-dimensional polycubes by R. Barequet, G. Barequet, G Rote.