Compute the following sum $ \sum_{i=0}^{n} \binom{n}{i}(i+1)^{i-1}(n - i + 1) ^ {n - i - 1}$? I have the sum 
$$ \sum_{i=0}^{n} \binom{n}{i}\cdot (i+1)^{i-1}\cdot(n - i + 1) ^ {n - i - 1},$$
but I don't know how to compute it. It's not for a homework, it's for a graph theory problem that I try to solve.

Look up Abel's binomial theorem,
for example, here:
http://en.wikipedia.org/wiki/Abel's_binomial_theorem
I find this funny because,
immediately preceding this,
I answered a question
by giving a reference to
Abel's summation formula.
This means,
of course,
that somebody will ask
about solving the quintic.
