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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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It's zero if $n$ is not a multiple of 4. Have you run any experiments? like, calculating $P(n)$ for a few small values of $n$? – Gerry Myerson Aug 24 '12 at 23:45
In this case, I mean integer division. I care about this in cases when n is large. – Paul S. Boudreaux Aug 24 '12 at 23:55
If you mean integer division, you should edit your question so that it says what you mean instead of saying something different. And small values often give insight into large ones. – Gerry Myerson Aug 25 '12 at 0:02
So one is meant to interpret \ as a symbol for integer division? Is that a standard notation? – Gerry Myerson Aug 25 '12 at 0:09
If $n\equiv 0\pmod 4$, it is the constant term of the expression $$\prod_{i=1}^n (xz^{i}+x^{-1}z^{-i})$$ Not sure if that helps any – Thomas Andrews Aug 25 '12 at 0:13

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.

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