The TAs in my department are stuck in assisting an undergraduate with the following problem:
$\sum^{2k}_{i=0} C^{4k}_{2i}(-1)^{i}=2^{2k}(-1)^{k}.$
We tried to solve this via induction (obviously failed), via various combinatorial identities, via generating functions, etc. Aside from the fact that nothing works, we also do not know how to solve this nicely by using some combinatorial interpretation given this is some undergraudate's HW. I venture to ask in here for I want to see how it can be properly understood.