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.



Is there a natural relationship between OEIS A127670 and Cayley's tree formula?
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](http://page.mi.fu-berlin.de/rote/Papers/pdf/Formulae+and+growth+rates+of+high-dimensional+polycubes.pdf) by R. Barequet, G. Barequet, G Rote.


