Euler's famous pentagonal number theorem is somewhat like this problem, except it deals with the generating function for partitions into distinct parts:
(1+x)(1+x^2)(1+x^3)...
If you change *all* of the + signs in the above into minus signs, then the statement of your conjecture holds; indeed there is an explicit formula for the terms of the generating function involving the pentagonal numbers, hence the name of the theorem. This theorem has several pretty and well-publicized proofs (see Chapter 1 of the introduction to "The Theory of Partitions" by George Andrews, or Chapter 14.5 of "Introduction to Analytic Number Theory" by Tom Apostol, or "Proofs from the book" by Aigner-Ziegler, or Wikipedia).
I would wager that this observation isn't terribly helpful, but still. Was this was the motivation of the problem?
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pentagonal number theorem
Euler's famous pentagonal number theorem is somewhat like this problem, except it deals with the generating function for partitions into distinct parts:
(1+x)(1+x^2)(1+x^3)...
If you change *all* of the + signs in the above into minus signs, then the statement of your conjecture holds; indeed there is an explicit formula for the terms of the generating function involving the pentagonal numbers, hence the name of the theorem. This theorem has several pretty and well-publicized proofs (see Chapter 1 of the introduction to "The Theory of Partitions" by George Andrews, or Chapter 14.5 of "Introduction to Analytic Number Theory" by Tom Apostol, or "Proofs from the book" by Aigner-Ziegler, or Wikipedia).
I would wager that this observation isn't terribly helpful, but still. Was this was the motivation of the problem?