$$\sum_{n=1}^\infty (-1)^n=-1+1-1+1-1+1\pm\cdots$$ $$\sum_{n=1}^\infty (-1)^{n(n+1)/2}=-1-1+1+1-1-1+1+1\pm\cdots$$ $$\sum_{n=1}^\infty (-1)^{n(n+1)(n+2)(n+3)/8}=-1-1-1-1+1+1+1+1\pm\cdots$$
Generally speaking, what is the series notation for a series following the above pattern, with $m$ negative signs followed by $m$ positive signs, etc.? They seem to follow the pattern $\sum_{n=1}^\infty (-1)^{n(n+1)\cdots(n+m-1)/2^{m-1}}$, but that falls apart when you consider $m=3$, where
$$\sum_{n=1}^\infty (-1)^{n(n+1)(n+2)/2^{2}}\neq-1-1-1+1+1+1\pm\cdots$$
Is there a general formula that, for some $m$, will give the correct series notation, regardless of the value of $m$?
I would prefer a notation of the form $\sum_{n=1}^\infty (-1)^{n(n+1)\cdots/x}$ if possible, but if that isn’t going to work, then go ahead and post some other function. I will also accept some rework of this function that starts with positives instead of negatives, as long as the general idea of alternating sets of positives and negatives remains the same.