Compute telescopic sum of binomial coefficients Is there a nice or simple form for a sum of the following form? 
$$ 1 + \sum_{i=1}^k \binom{n-1+i}{i} - \binom{n-1+i}{i-1}$$
Motivation: Due to a computation in the formalism of Schubert calculus the above sum with $k = \lceil n/2 \rceil -1$  is equal to the number of lines intersecting $2n-4$ general subspaces $H_j\subseteq \mathbb{P}^n$ of dimension $n-2$.

No need for a power tool when a manual tool does the trick.
Hockey-stick identity:
$$
\sum_{i=0}^k \binom{n+i}{i} = \binom{n+k+1}{k}
$$
Applying to our expression:
$$
1 + \sum_{i=1}^k \binom{n-1+i}{i} - \binom{n-1+i}{i-1} \\
1 + \sum_{i=1}^k \binom{n-1+i}{i} - \sum_{i=1}^k \binom{n-1+i}{i-1} \\
1 - \binom{n-1}{0} + \sum_{i=0}^k \binom{n-1+i}{i} - \sum_{i=0}^k \binom{n+i}{i} \\
\sum_{i=0}^k \binom{n-1+i}{i} - \sum_{i=0}^k \binom{n+i}{i} \\
\binom{n+k}{k} - \binom{n+k+1}{k} \\
\binom{n+k}{k} - \left(\binom{n+k}{k-1}+\binom{n+k}{k}\right) \\
-\binom{n+k}{k-1}
$$
