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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 $2^{n/2}$?

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    (continued) $\Omega(\frac{2^n}{n^{2.5}})$, in particular larger than $2^{\frac{n}{2}}$ for large n.2012-08-25

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For $n=4k$, this is OEIS sequence A063074. That entry and the one for A029895 that it links to contain some suggestions for asymptotic expressions derived heuristically and/or experimentally, but no closed form or proof for an asymptotic expression seems to be known.