I've been exploring a few generating function proofs to show that two sets have the same number of elements, and a common tactic is to divide the generating functions by each other, and obtain one. Twice so far, this has occurred with me by an infinite cascade of different of two squares.
For example, to prove that the number of ways to write a number as a sum of powers of 2 is unique, we can consider its generating function $(1+x)(1+x^2)(1+x^4)...$ Dividing that for the generating function which we are targetting $1/(1-x)$, and we get $(1-x)*(1+x)*(1+x^2)$... It is clear that any power of x, other than 0, has a coefficient of 0, as the cancellation of terms by the difference of two squares is never ending.
I am fully aware of the combinatoric proof of that fact; that is not what my question about. My question is specifically about multiplying by $(1-x)$. This is naturally the reciprocal of $1/(1-x)$, so it is like dividing by it, but there is no combinatoric meaning to dividing generating functions. Furthermore, I know that multiplying by $1/(1-x)$ is combinatoric equivalent to getting the partial sums of a sequence, so multiplying by $1-x$ is the "inverse" of that operation, the partial differences. That is not what I am looking for. What I want is a way to combinatorically interpret the term $1-x$ independently of its reciprocal if possible.
For example, if my question were about $(1+x)$ then a satisfactory answer would be its relation to the binomial coefficients, or more broadly, how the new term's coefficient counts the number of elements in the union of the old coefficient and its precedent. ($[x^n]+[x^{n-1}]$
A similar interpretation for $(1-x)$ would be that the new term's coefficient counts the number of elements in the difference of the old coefficient and its preceding coefficient? Does this even make sense unless the preceding coefficient represents the size of a subset of the old coefficient itself?
I'm really just trying to understand how this generating function proof is correlated to a combinatoric proof. Like, what is the significance of multiplying $(1-x)$ and what do these cascading cancellations really mean combinatorically?
Btw, I've seen the "cascading cancellation" when proving a partition related identity; and it also came in the context of unique representation: "every number has a unique way of being written as the product of an odd number and a power of two". So must there be a relation between this cancellation process and uniqueness of representation of a number in a particular manner?