@GjergjiZaimi Will similar arguments work if f(n) is taken to be 1+x^n+x^2n+x^3n+...+x^kn . ?

@Fedor Petrov I've made a correction in response to your comment , but I'm not sure if it answers your question.

@MaxAlekseyev Thanks for the code. Unfortunately I've been unable to follow it thusfar, perhaps because I'm unfamiliar with Pari. I've been using Mathematica only for the last several years.

@MaxAlekseyev Thank you for this. How long did your search for the maximal series take? Could you provide me with the program? Your solution with degree 121 is better than I ever found. The next logical thing to try by way of a search is to write 1/(1-x^2) as 1+x^2+x^4+...before changing the signs.

I've now read the revised version of the answer by Aaron Meyerowitz and checked one of the largest degree maximal polynomials. This case of my old problem seems to be solved. Does anyone think they can solve the next case? By this I mean: start with the generating function for unrestricted partitions as above and change signs in the first two expressions in parentheses of the product. Can the new series have coefficients all of which are 1, -1, and zero? This is much harder.

I've now read the revised version of the answer by Aaron Meyerowitz and checked one of the largest degree maximal polynomials. This case of my old problem seems to be solved. Does anyone think they can solve the next case. By this I mean: s

I'm glad to see that the branch that you've chosen here is in agreement with what I've calculated. It is impossible to continue this branch to 25. What you are indicating with the c's and d I do not understand. All that would be necessary to do is continue the same sort of calculations for all the other branches. I think I have accomplished this but it would be nice to have independent confirmation. Otherwise I'll just do the computations again by myself.

@Alexey Ustinov I just noticed that when you edited the problem you left off the ellipsis in the term that begins 1+x^3+x^6+x^9. There should be a "+..." there.

@Alexey Ustinov. Ultimately, my motivation for this problem came from a comparison of the generating function for partitions into distinct parts (1+x)(1+x^2)(1+x^3)... with the series (1-x)(1-x^2)(1-x^3)...=1-x-x^2+x^5+x^7-x^12-... I wanted to find other instances where a change of sign led to some pretty result.