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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?

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?

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.

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?

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?

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+...

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?

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.

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?