Identity with binomials Does there exist a closed formula for $$\underset{n=1}{\overset{N-1}{\sum}}\dbinom{N+n}{n}?$$ I've searching on wikipedia but I haven't found this kind of sum.

Yes, this sum is equal to
$$ \frac{N}{N+1}{2N \choose N}-1.$$
Proof: Consider a more general identity
$$\sum_{n=0}^{M-1}{N+n \choose n} =\frac{M}{N+1}{N+M \choose M}, $$
which can be proved by induction on $M$.
