Coefficients of a relative of cyclotomic polynomials

Let $N=p_1{p}_2\mathrm{.}\mathrm{.}\mathrm{.}{p}_n$ be a product of $n$ distinct primes. Define $P_N(x)$ to be the polynomial $(1-x^N)\prod_{1\leq i<j\leq n}(1-x^{N/(p_ip_j)})/\prod_{i=1}^n (1-x^{N/p_i})$\mathrm{.}(When$n=2$,$P_{pq}(x)$ is the $pq$-th cyclotomic polynomial, and when $n=3$, $P_{pqr}(x)$ is $(1-x)$ times the $pqr$-th cyclotomic polynomial.) Let the height of a polynomial be the maximum absolute value of one of its coefficients. It is well known that the height of ${\mathrm{\Phi }}_{pq}(x)$ is 1, and Gallot and Moree showed that the same is true for $P_{pqr}(x)$ when $n=3$. We show that the coefficients of $P_N(x)$ depend mainly on the relative order of sums of residues of the form $p_j^{-1} \pmod {p_i}$\mathrm{.}Thisallowsustoexplicitlydescribethecoefficientsof$P_N(x)$when$n=3$andshowthattheheightof$P_N(x)$isatmost2when$n=4$\mathrm{.}Wealsoshowthatforany$n$thereexist$P_N(x)$ with height 1 but that in general the maximum height of $P_N(x)$ is a function depending only on $n$ with growth rate $2^{n^2\mathrm{/}2+O(n\log n)}$.