Let's consider the $1$-variable rational function $$F(z):=\frac{1-z}{(z^3 - z^2 + 2z - 1)\,(z^3 + z^2 + z - 1)}.$$

Numerical evidence convinces me of the truth of the following.

QUESTION. Can you prove that $F(z)$ is positive, in the sense that its Taylor series at $z=0$ has positive coefficients?

Note. I'm not sure whether the concept of multi-sections of a series is efficient for the present purpose.

Let's consider the $1$-variable rational function $$F(z):=\frac{1-z}{(z^3 - z^2 + 2z - 1)\,(z^3 + z^2 + z - 1)}.$$

Numerical evidence convinces me of the truth of the following.

QUESTION. Can you prove that $F(z)$ is positive, in the sense that its Taylor series at $z=0$ has positive coefficients?

Note. I'm not sure whether the concept of multi-sections of a series is efficient for the present purpose. Nor do I think that looking at asymptotic growth of largest positive real roots is any more elegant.