  Let $G$ be the fundamental group of a compact nonpositively curved cube
complex $Y$. With respect to a basepoint $x$, one obtains an integer-valued
length function on $G$ by counting the number of edges in a minimal length
edge-path representing each group element. The growth series of $G$ with
respect to $x$ is then defined to be the power series $G_x(t)=\sum_g t^{|g|}$
where $|g|$ denotes the length of $g$. Using the fact that $G$ admits a
suitable automatic structure, $G_x(t)$ can be shown to be a rational function.
We prove that if $Y$ is a manifold of dimension $n$, then this rational
function satisfies the reciprocity formula $G_x(t^{-1})=(-1)^n G_x(t)$. We
prove the formula in a more general setting, replacing the group with the
fundamental groupoid, replacing the growth series with the characteristic
series for a suitable regular language, and only assuming $Y$ is Eulerian.
