Asymptotics – fine-tuning the growth rate of polynomial degrees

To let $ r $ be an integer with $ r> 1 $, Suppose that $ p_ {k} (x) $ is a polynomial with positive integer coefficients with $ p_ {k} (0) = 1 $ but where $ p_ {k} neq 1 $ for all $ k geq 0 $,

Suppose that
$$ prod_ {k = 0} ^ { infty} p_ {k} (x) = frac {1} {1-rx}

For each $ n> 0 $, To let $ t_ {n} $ be the number indexes $ k $ from where $ deg (p_ {k} (x)) = n $, Then polynomials can be selected $ (p_ {k}) _ {k geq 0} $ from where
$$ | t_ {n} – frac {r ^ {n}} {n} | = O ( alpha ^ {n}) $$
for each $ alpha> 1 $?

How slow can this work? $ n mapsto | t_ {n} – frac {r ^ {n}} {n} | $ to grow? How slow can this work? $ n mapsto max (0, frac {r ^ {n}} {n} -t_ {n}) $ to grow? For example we can have $ max (0, frac {r ^ {n}} {n} -t_ {n}) = O ( alpha ^ {n}) $ for all $ alpha> 1 $?

This question is motivated by very great cardinals.