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Ira Gessel
Ira Gessel
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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 $$b_n = n^{n-2}$$,$$b_n = n^{n-2},$$ which means that $$a_n = \frac{2^n}{n+1} b_{n+1}$$.$$a_n = \frac{2^n}{n+1} b_{n+1}.$$ 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.

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.

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.

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Gjergji Zaimi
Gjergji Zaimi
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Tom Solberg
Tom Solberg
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Is there a natural relationship between OEIS A127670 and Cayley's tree formula?

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.