This is my latest attempt to simplify an old problem of mine so much that the simplified problem can actually be answered.

Starting with the generating function for unrestricted partitions:

(1+x+x^2+x^3+...)(1+x^2+x^4+x^6+...)(1+x^3+x^6+x^9)...

Change some of the plus signs in the leftmost expression in parentheses to minus signs. Is it possible that the resulting series has coefficients all of which are 1, -1 ,or zero?

I believe that the answer is no, but I'm not convinced by my computer aided search.