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Let $r$ be an integer with $r>1$. Suppose that if $k\geq 0$, then $p_{k}(x)$ is a polynomial with nonnegative integer coefficients with $p_{k}(0)=1$ but where $p_{k}\neq 1$.

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

For each $n>0$, let $t_{n}$ be the number indices $k$ where $\deg(p_{k}(x))=n$. Then is it possible to select polynomials $(p_{k})_{k\geq 0}$ where $$|t_{n}-\frac{r^{n}}{n}|=O(\alpha^{n})$$ for each $\alpha>1$?

How slowly can the function $n\mapsto|t_{n}-\frac{r^{n}}{n}|$ grow? How slowly can the function $n\mapsto\max(0,\frac{r^{n}}{n}-t_{n})$ grow? For example, can we have $\max(0,\frac{r^{n}}{n}-t_{n})=O(\alpha^{n})$ for all $\alpha>1$?

This question is motivated by very large cardinals.

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    $\begingroup$ It's not so hard to get $\alpha = \sqrt{r}$, or, for the second version, $r^{1/3}$. $\endgroup$ – Will Sawin Feb 14 at 3:57

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