This is a finite version of a problem of mine entitled "A product related to unrestricted partitions." It has the advantage that, at least for small values of n, it is easily solved.

Begin with the generating function for partitions with largest part n:

x$^n$ (1+x+x$^2$+...)(1+x$^2$+x$^4$+...)(1+x$^3$+x$^6$+...)... (1+x$^n$+x$^{2n}$+x$^{3n}$...)

Now replace some of the plus signs with minus signs. Expand the product. Is it possible that the resulting series has coefficients which all belong to the set {-1,0,1}?

For n=1,2,..., 8 the answer is yes. n=9 is the first value for which I don't know the answer.

Here's an example of a solution for n=4: (1+x+x$^2$+x$^3$+...)(1-x$^2$+x$^4$-x$^6$+...)(1-x$^3$+x$^6$-x$^9$+...)(1-x$^4$+x$^8$-x$^{12}$+...) The signs in the first factor are all +, while the signs in the second, third, and fourth factors alternate: +-+-... The resulting series has coefficients which are periodic with period 24.


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