# Coefficients of some infinite product power series

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 $$|a_k|$$?