Start with the product

$$(1+x+x^2) (1+x^2)(1+x^3)(1+x^4)\cdots$$

(The first polynomial is a trinomial..The others are binomials..) Is it possible by changing some of the signs to get a series all of whose coefficients are $ -1,0,$or $1$?

A simple computer search should suffice to answer the question if the answer is "no." I haven't yet done such a search myself.

This question is a takeoff on the well known partition identities like:

$$\prod_{n=1}^{\infty} (1-x^n)= 1-x-x^2+x^5+x^7-\ldots$$

"What for? Where could that be applied?"What's the point of asking this question, especially in this particular form, all over the site? If you don't like the post or it is not interesting to you, just ignore it. The problem is well-posed and the answer is not immediately obvious. What else do you want from a mathematical question? Somebody is interested in it for some reason and he has no more obligation to explain to you why than you have to explain to him what the meaning and the purpose of your life are. Just live and let live :-) $\endgroup$ – fedja Jun 9 at 1:11