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](https://mathoverflow.net/questions/74191/what-is-the-degree-of-a-symmetric-boolean-function) 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](https://math.stackexchange.com/questions/3445254/non-trivial-alternating-sums-of-binomial-coefficients) but then I read that this community if better suited for research-level questions.]



Non-trivial alternating sums of binomial coefficients
(Reposting my comment as an answer, as requested by the OP.)


If you have a solution with $a\_i \in \{−1,1\}$, then you also have a solution with $a\_i \in \{0,1\}$ simply by replacing each $a\_i$ with $(a\_i+1)/2$ (using the fact that setting all $a\_i=1$ is also a valid solution). In other words, your question had already been answered by some of the comments to [this question](https://mathoverflow.net/questions/74191/what-is-the-degree-of-a-symmetric-boolean-function).


