If we define $x^n=\overbrace{x\star x\star\cdots\star x}^{n\text{ copies of }x},\; a\star b=-(a+b)$, where the final expression is evaluated with the usual definitions of $+$ and $-$, how many possible values are there for $x^n$, where the expression can be parenthesized in any way?
For $n\in\{1,2,3\}$, it's just one ($x$, $-2x$, $x$, respectively), but when $n=4$, there are two ($-2x$ and $4x$). I don't even know what to search for this, so that's why I'm asking here.