Let $C_n$ denote the $1$-skeleton of the $n$-dimensional cross-polytope, and $\chi_{C_n}(x)$ be the chromatic polynomial of $C_n$. This is equivalent to the way of coloring the $(n-1)$-dimensional faces of the $n$-dimensional hypercube with $x$ colors so that if two of them share a $(n-2)$-dimensional face, they have different colors.
Then
- $\chi_{C_1}(x) = x^2,$
- $\chi_{C_2}(x) = x(x-1)(x^2-3x+3),$
- $\chi_{C_3}(x) = x(x-1)(x-2)(x^3 - 9x^2 + 29x - 32),$
- $\chi_{C_4}(x) = x(x-1)(x-2)(x-3)(x^4 - 18 x^3 + 125 x^2 - 392 x + 465),$
- $\chi_{C_5}(x) = x(x - 1) (x - 2)(x - 3) (x - 4)(x^5 - 30 x^4 + 365 x^3 - 2240 x^2 + 6909 x - 8544),$ and more generally,
- $\displaystyle \chi_{C_n}(x) = \left(\prod_{i=0}^{n-1}(x-i)\right)f_n(x)$.
It appears that the absolute value of the constant term of $f_n(x)$ is equal to OEIS sequence $A007677(3n - 4)$ for $n \geq 2$, where A007677 is a list of "denominators of convergents to $e$".
Is this a coincidence, or is there a connection between hypercubes and convergents of $e$?
I'm surprised by this. Should I be?