Is there a proof that $(n-2x)! \times n^{2x-1} > n!$ (where $x$ is a function related to the prime counting function

Question

Is it possible to prove the following?

Let $\pi$ be the prime counting function and $A(n)=\pi(2n)-\pi(n)$

$(n-2A(n))! \times n^{2A(n)-1} > n!$