Let $f(n)\colon \mathbb{P}\to\mathbb{R}_{>0}$, where $\mathbb{P}=\{1,2,\dots\}$, be some ''nice'' function such that $f(n) \to \infty$ as $n\to\infty$. For instance, $f(n)=1+\log(n)$ or $f(n)=n$. Let $$ \prod_{i\geq 1}(1-x^i)^{f(i)} = \sum_{k\geq 0} a_kx^k. $$ What can be said about the coefficients $a_k$? In particular, let $b_k$ be the length of the $k$th maximal sequence of consecutive same-sign elements of the sequence $a_0,a_1,a_2, \dots$ (where when $a_k=0$ we consider the sign to be positive). For instance, when $f(n)=n$ we have $$ \prod_{i\geq 1}(1-x^i)^i = 1-x-2x^2-x^3+4x^5+4x^6+7x^7 +3x^8-2x^9-\cdots, $$ so $b_1=1$, $b_2=3$, and $b_3=5$. Is it true that $b_k\to\infty$ as $k\to\infty$? In some random cases I checked, it appears that if $f(n)=O(n^c)$ for some $c>0$, then for $k$ sufficiently large we have $|b_k-b_{k+1}|\leq 1$. For instance, for $f(n)=n$ we have $$ (b_1,b_2,\dots) = (1,3,5,7,7,8,8,10,10,11,11,12,13,13,13,15,14,15, 15,\dots), $$ and we have $|b_k-b_{k+1}|\leq 1$ for $16\leq k\leq 256$.

Can we say more about the rate of growth of $b_k$? For instance, for $f(n)=n$ it looks like $b_k$ is approximately equal to $k^{3/4}$. What can we say about the $a_k$'s? Is it true that $|a_k|\to\infty$ as $k\to\infty$? Can we say something about the rate of growth of $|a_k|$?


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