# The number of ways to represent $(n^2+n)/4$ as a sum of $n/2$ distinct integers in $1,\dots,n$

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 $$2^{n/2}$$?

• 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

## 1 Answer

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.