Consider a binary vector $a_0, a_1,\,\dots\,, a_n$ and an equation

$$\sum_{i=0}^n a_i \cdot (-1)^i {n \choose i} = 0.$$

You can satisfy this trivially when

1) all $a_i$ are 0, or

2) all $a_i$ are 1, or

3) $n$ is odd and $a$ satisfies $a_i = a_{n-i}$ for all $i$, because the summands for $i$ and $n-i$ cancel out.

My question is if there are any other vectors $a$ satisfying the equation?

[There has been a related question but regarding $a_i \in \{-1,1\}$. There are non-trivial examples given in the answers but they do not seem to work in this setup. I have also asked this at math.stackexchange but then I read that this community if better suited for research-level questions.]

binaryvector, which I think means a vector whose entries are just zeros & ones. $\endgroup$LeechLattice) by reflection and 1's complement; the twelve for $n=14$ are the orbits of $000001001110000, 000001100110000, 000010000110000$. $\endgroup$