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][1] 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][2] but then I read that this community if better suited for research-level questions.]


  [1]: https://mathoverflow.net/questions/74191/what-is-the-degree-of-a-symmetric-boolean-function
  [2]: https://math.stackexchange.com/posts/3445254/edit