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Given two integers $n$ and $N$ such that $N>{n+1\choose 2}$, we denote by $\alpha_n(N)$ the number of elements $(x_1,\dots,x_n)$ in $(\mathbb Z/N\mathbb Z)^n$ such that the $2n$ elements $x_1,x_1+1,x_2,x_2+2,x_3,x_3+3,\dots,x_n,x_n+n$ of $\mathbb Z/N\mathbb Z$ are all distinct. The number $\alpha_n(N)$ is then a polynomial function of the form $N^n-2nN^{n-1}+\dots$.

Is there a nice formula for the polynomial $\alpha_n(N)$?

Remark: The analogous problem such that $x_1,x_1+1,x_2,x_2+2,x_3,x_3+4,x_4,x_4+8,\dots, x_n,x_n+2^{n-1}$ (and many similar variations and generalizations) is much easier and I have a solution (described in

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