TITLE: The number of positive integer solutions to the equation $x_1+2x_2+...+nx_n=n^2.$
QUESTION [6 upvotes]: Let $n \ge 2, n \in \mathbb N$. $A_n$ denotes the number of positive integer solutions to the equation
  $$x_1+2x_2+...+nx_n=n^2.$$
  Prove inequality
  $$\frac{n^n(n-1)^{n-1}}{2^{n-1}\left(n!\right)^2}<A_n<\frac{n^{2n-1}}{\left(n!\right)^2}$$

I have no idea how to solve this problem.

REPLY [2 votes]: From the OP's question The number of positive integer solutions to the equation $x_1+x_2+...+x_n=n^2.$, we have 
$$A_n>\frac{1}{2^{n-1}n!}\,\binom{n^2-1}{n-1}\geq\frac{\big(n(n-1)\big)^{n-1}}{2^{n-1}n!(n-1)!}=\frac{n^n(n-1)^{n-1}}{2^{n-1}(n!)^2}\,.$$