Binomial coefficients with even interior terms ($\left(\genfrac{}{}{0ex}{}{N}{j}\right)\equiv 0(\mathrm{mod}2)(0<j<N)⟺N=2^k$)

For a positive integer $N$, all binomial coefficients $\left(\genfrac{}{}{0ex}{}{N}{j}\right)$ with $0<j<N$ are even exactly when $N$ is a power of two. Write the binary expansion $N={\sum }_{i\in I}{2}^i$. In the polynomial ring ${\mathbb{F}}_2[t]$, repeated squaring givesIf $I$ has one element, there are no interior terms. If it has more than one, the coefficient of $t^{2^{\min I}}$ is one and that exponent lies strictly between zero and $N$. This characterization gives a useful parity obstruction directly from the binary expansion.