Start with the product for unrestricted partitions: (1+x+x$^2$+...)(1+x$^2$+x$^4$+...)(1+x$^3$+x$^6$+...)...$(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...$
Now replace some of the plus signs with minus signs and expand the product into a series. Is it possible that the coefficients of the resulting series can all be in the set {+1,0,-1}?
Here's an example for which the coefficients of x$^n$ are from the required set for n<=9.
(1+x-x$^2$-x$^3$-x$^4$-x$^5$+x$^6$+x$^7$+x$^8$+x$^9$)
\
(1+x$^2$-x$^4$-x$^6$-x$^8$)
(1+x$^3$-x$^6$-x$^9$)
(1+x$^4$-x$^8$)(1+x$^5$)( 1+x$^6$)(1+x$^7$)(1+x$^8$)(1+x$^9$)
=1+x+x$^3$-x$^4$-x$^5$+x$^6$-x$^7$+...
This can be extended to a product which satisfies the requirements for n<65.
I've found examples where the coefficients are good for n exceeding 100, but not as large as 110.
Over the years I've asked this question of several mathematicians. Two of them ventured their opinions. Freeman Dyson thought it couldn't be done. George Andrews thought it couldn't be done in any meaningful way. (These are not exact quotes, just the sense of their answers.)
