I posed a question called "A Product Related to Unrestricted Partitions". As it stands it is too hard. Here's another variation which is easier to search for and hopefully might shed some light on the harder problem..

Begin with the generating function for unrestricted partitions written out as follows:

$$\frac{1+x+x^2+x^3+\dots}{(1-x^2)(1-x^3)(1-x^4)\cdots}$$

Now change some of the signs in these factors. Is it possible that the resulting series has coefficients all of which are +1, -1, or zero? Failing this, what is the longest solution with coefficients +1, -1, or zero that can be found?