How many integer solutions $(m_0,m_1,\ldots,m_n)$ to the double inequality $$n\sum\limits_{i=0}^n \frac{1}{m_i^2} > \left(\sum\limits_{i=0}^n \frac{1}{m_i}\right)^2 > (n-1)\sum\limits_{i=0}^n \frac{1}{m_i^2}\ ?$$
Is there an efficient algorithm for me to find (using a computer) the number of $n+1$-tuples $(m_0,...,m_n)\in\mathbb{N}^{n+1}$ with $M<m_i<N$ for fixed natural numbers $M,N$ satisfying the equation in the title?
I cannot count it one by one since $N$ and $n$ can be large.
Edit : It seems like in the general case so many tuples in $[M,N]^{n+1}$ satisfy the inequality. I would also appreciate some kind of a asymptotical answer i.e. the ratio of the number of solutions with $(N-M)^{n+1}$.