Alternate proof for weighted alternating shifted central binomial sum relation
Solution 1:
Sorry to bump an old question, but I thought this would interest you.
In this paper with Divesh Aggarwal, Huck Bennett, and Sasha Golovnev, we ended up using this identity and a generalization of it to prove certain computational hardness results for lattice problems. See Section 7 there.
Here's the full statement:
OP's statement corresponds to the case when $k = 2\tau$.
The proof uses the contour integration idea suggested by @tired.
Anyway, thanks for posting this! I doubt we would have discovered this identity without you!