Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$

The following equality is famous:

$$\cfrac{q^{1/5}}{R(q)} = \prod_{k>0} \cfrac{(1-q^{5k-2})(1-q^{5k-3})}{(1-q^{5k-1})(1-q^{5k-4})} ( = f(q))$$

The coefficients of $f(q)$ can be positive or negative. In fact,

$$f(q) = 1 + q - q^3 + q^5 + q^6 - q^7 - 2 q^8 + \cdots$$

Let

$$g(q) = \prod_{k>0} f(q^k) = f(q)f(q^2)f(q^3) \cdots$$

$g(q)$

$= (1 + q - q^3 + q^5 + \cdots)(1 + q^2 - q^6 + q^{10} + \cdots)\cdots$

$= 1 + q + q^2 + q^3 + 2q^4 + 3q^5 + 3q^6 + 3q^7 + 4q^8 + 6q^9 + \cdots$

The coefficients of $g(q)$ seem non-negative. Are the coefficients of $g(q)$ non-negative?