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.]

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.]