Value distribution of cyclotomic polynomial coefficients

Let $a_n(k)$ be the $k$th coefficient of the $n$th cyclotomic polynomial $\Phi_n(x)$. As $n$ ranges over the integers, $a_n(k)$ assumes only finitely many values. For any such value $v$ we determine the density of integers $n$ such that $a_n(k)=v$. Also we study the average of the $a_n(k)$. We derive analogous results for the $k$th Taylor coefficient of $1/\Phi_n(x)$ (taken around $x=0$). We formulate various open problems.