Prove that for every $n\in \mathbb{N+}$ the following equation is not satisfied:
${{n}\choose {1}}-2\cdot{{n}\choose {3}}+4\cdot {{n}\choose {5}}-8\cdot {{n}\choose {7}}+...\pm2^{\frac{k-1}{2}}\cdot{{n}\choose {k}} = 0$,
where $k$ is the largest odd number not greater than $n$.
I think it's we need to consider 4 cases ($n$ modulo $4$), but I don't know how to finish the proof.