Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $2^{n/2}$?

share|improve this question
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
2  
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
2  
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 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.