KöMaL 2022 A830

For $H\subset \mathbb{Z}$ and $n\in \mathbb{Z}$, let $h_n$ denote the number of finite subsets of $H$ in which the sum of the elements is $n$. Does there exist $H\subset \mathbb{Z}$, for which $0\notin H$, and $h_n$ is a (finite) even number for every $n\in \mathbb{Z}$? (The sum of the elements of the empty set is 0.)