# Sequences and series – factorization of \$ frac {1} {1-rx} \$ into an infinite product of polynomials

I'm looking for examples of polynomial sequences $$(p_ {k} (x)) _ {k = 1} ^ { infty}$$ with positive integer coefficients where $$p_ {k} (0) = 1$$ for all $$k geq 1$$ and where is there a positive integer? $$r$$ from where
$$frac {1} {1-rx} = prod_ {k = 1} ^ { infty} p_ {k} (x)$$
whenever $$x in (0, frac {1} {r})$$,

It is easy to construct many examples of polynomials $$p_ {k} (x)$$ which fulfill this infinite product formula by recursion. Let us limit the scope of this question to sequences $$(p_ {k} (x)) _ {k = 1} ^ { infty}$$ which have a closed expression (unless there is a very good reason for the deviation) and are sequences in which Srinivasa Ramanujan may be interested.

Ex 1: $$frac {1} {1-x} = prod_ {k = 1} ^ { infty} (1 + x ^ {2 ^ {k}})$$
Ex 2: $$frac {1} {1-2x} = prod_ {k = 1} ^ {k} (1 + x ^ {k}) ^ {a (k)}$$
from where $$a (k)$$ is sequence A-306156 in the OEIS. Similar products are available for $$frac {1} {1-rx}$$ for all $$r> 1$$,