For any positive integer $n$ (using integer division only), let $P(n)$ denote the number of ways in which the integer $(n^2+n)/4$ can be expressed as a sum of exactly $n /2$ distinct elements of the set $\{1,2,3,\dots, n\}$.
What is $P(n)$ in terms of n? Specifically, how exponential is it? Is this less than $2^{n/2}$?