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Consider the two series defined by

series 1: $$(1+x+x^2+x^3+...) (1+x^3+x^6+x^9...)(1+x^5+x^{10}+x^{15}+...)...$$ and series 2: $$1+x+x^2+x^5+x^7+...$$ Where the exponents 1,2,5,7,...are the pentagonal numbers of the form $k(3k-1)/2$.

By changing some of the signs of the coefficients one may make the two series equal. I know of 4 ways to do this. See the paper Integers 18 (2018) George Andrews and David Newman, Binary Representations and Theta Functions.

Are there any other changes of sign, aside from these four which can make the two series equal?

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    $\begingroup$ It would be nice if these known changes of sign are included, so as to make the question self-contained. $\endgroup$
    – YCor
    Commented Jul 4, 2019 at 18:20
  • $\begingroup$ I agree that it would helpful to make "these four" explicit. The article is Integers 18 A34, math.colgate.edu/~integers/s34/s34.pdf, which I believe includes just one change of sign example for this situation, equation (4.1). $\endgroup$
    – Brian Hopkins
    Commented Jul 17, 2019 at 4:39

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