I am interested in any information about coefficients of the polynomials $\Psi_n(x):=\prod_{k=1}^n\Phi_k(x)$, where $\Phi_k$ are the cyclotomic polynomials.
I realize of course that this is a complicated object and presumably very little is known about it, but even that little I don't know.
Practically the only thing I know is the degree - it is $D(n):=\sum_{k=1}^n\varphi(k)$, where $\varphi(k)$ is Euler's totient function. This is easy to show anyway, the degree coincides with the number of elements in the $n$th Farey sequence, as the polynomial has all primitive roots of 1 up to $n$ as its simple roots.
To pose some concrete (sub)question - it seems that one has $$ \Psi_n(e^{it})=2ie^{\frac{D(n)}2it}\sum_{k=1}^{\frac{D(n)}2}c_n\left(k\right)\sin(kt) $$ where, for each $n$, $c_n\left(k\right)$, $k=1,...,\frac{D(n)}2$ is a unimodal sequence of natural numbers - for example,
- $2ie^{2it}(\sin(t)+\sin(2t))$
- $2ie^{3it}(\sin(t)+\sin(2t)+\sin(3t))$
- $2ie^{5it}(2\sin(t)+3\sin(2t)+3\sin(3t)+2\sin(4t)+\sin(5t))$
- $2ie^{6it}(\sin(t)+2\sin(2t)+2\sin(3t)+2\sin(4t)+\sin(5t)+\sin(6t))$
- $2ie^{9it}(4\sin(t)+7\sin(2t)+9\sin(3t)+9\sin(4t)+8\sin(5t)+6\sin(6t)+4\sin(7t)$ $+2\sin(8t)+\sin(9t))$
- $2ie^{11it}(5\sin(t)+9\sin(2t)+12\sin(3t)+13\sin(4t)+13\sin(5t)+11\sin(6t)+9\sin(7t)+6\sin(8t)+4\sin(9t)+2\sin(10t)+\sin(11t))$
Is there a chance to prove this unimodality, or just positivity, or any arithmetic properties of these coefficients? Like, what is $c_n\left(1\right)$? Or, it is more or less clear that the highest coefficient $c_n\left(\frac{D(n)}2\right)$ is 1; but what is the previous one, $c_n\left(\frac{D(n)}2-1\right)$?