A finite sum over $\pm 1$ vectors $$\mbox{What is a nice way to show}\quad
\sum_{\vphantom{\Large A}u\ \in\ \left\{-1,+1\right\}^{N}}
\left\vert\,\sum_{i\ =\ 1}^{N}u_{i}\,\right\vert
= N{N \choose N/2}
\quad\mbox{when}\  N\ \mbox{is even ?.}
$$
Could there be a short inductive proof ?.

Here is a recurrence relation (I doubt it will be of much use, but maybe it can help you in some way.
$s_{n+2}=2s_n+\sum_{i=0}^{N-2}\binom{N-2}{i}(|2i-N+2+2|+|2i-N+2-2|)$
It works by noting that for every sequence in $\{1,-1\}^{N-2}$ there are two ways of extending it without affecting the sum (by adding $-1,1$ or by adding $1,-1$), this gives the $2s_n$ term. There is one way to extend it by appending $2$ $1$s, this give the $\sum_{i=0}^{N-2}\binom{N-2}{i}|2i-N+2+2|$ term. And there is one way to extend it by appending $2$ $-1$s, this gives the $\sum_{i=0}^{N-2}\binom{N-2}{i}|2i-N+2-2|$ term.
Again, i doubt it can be of much use. And even if you can somehow use a prove by induction from here I'm pretty sure it will be longer than the answer provided by Aryabhata.
