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Prove that $\binom{n} k$ is even for all $k$ between $1$ and $n-1$, if and only if $n$ is a power of $2$.

I have tried strong induction on Pascal's triangle but I've got nowhere. I think that there is a generating function approach to this question that I'd like to find.

edited Nov 28 '16 at 17:08

Thomas Andrews

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asked Nov 28 '16 at 17:05

grigori

Have you shown either direction? It might be easy to show it in one direction first. – Thomas Andrews Nov 28 '16 at 17:10
This answer to another instance of essentially this question sketches a proof by induction on Pascal’s triangle. – Brian M. Scott Nov 28 '16 at 17:21

Prove that n choose k is even for 0

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