Begin by writing the generating function for unrestricted partitions as follows:
(1+x+x^2+x^3+…)(1+x^2+x^4+x^6+...)(1+x^3+x^6+x^9)⋯= 1+p(1)x+p(2)x^2+...
Now change some of the coefficients from plus to minus. In the resulting series what is the smallest that the coefficient of x^n can be? (Don't change the sign of any of the ones.) By smallest I mean that it is negative and that the absolute value is as large as possible. For odd values of the exponent n one can get a coefficient of -p(n). What happens for even n?
For example: Changing some of the first few coefficients as follows
(1 - x - x^2 + x^3 - x^4 - x^5 + x^6 - x^7 - x^8) (1 - x^2 + x^4 + x^6 - x^8) (1 + x^3 + x^6) (1 + x^4 - x^8) (1 - x^5) (1 + x^6) (1 + x^7) (1 - x^8) gives 1-x-2x^2+3x^3+x^4-7x^5+5x^6+5x^7-18x^8+...
The coefficient of x^8 is -18 and this is the smallest possible; p(8)=22 and -22 cannot be obtained by any choice of signs.
