It is possible. Select positive signs for all factors but the first. The product of all these factors is then the generating function for partitions without 1's. Since this sequence is sub-doubling, G Pasemans' answer to question Will this greedy algorithm always work? shows that we can select the signs in the first factor so as to make the coefficient $a(n)$ be $1$, $-1$ or $0$.
moshe newman
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