Studying a specific quantum cluster algebra, I have come across the following sequence of polynomials :

$$X_1$$ $$q^{-1/2}(X_1X_2-1)$$ $$q^{-1/2}(X_1X_2X_3-X_3-X_1)$$ $$q^{-1}(X_1X_2X_3X_4-X_3X_4-X_1X_2-X_1X_4+1)$$ $$q^{-1}(X_1X_2X_3X_4X_5 - X_1X_4X_5 - X_1X_2X_3 - X_1X_2X_5 - X_3X_4X_5 + X_1 + X_3 + X_5)$$

Either finding a simple expression, or just other places where these polynomials occur would be interesting. They differ only from a $q$ factor from their commutative analogues, which can be caracterized as determinants of the following matrix:

$$\begin{pmatrix} x_n & 1 & 0 & \cdots & 0 \\\\ 1 & x_{n-1} & 1 & \cdots & 0\\\\ 0 & 1 & x_{n-2} & \cdots & 0\\\\ \vdots & \vdots & \vdots & \ddots & \vdots \\\\ 0 & 0 & 0 & \cdots & x_1 \end{pmatrix}$$

I read about a quantum analog of determinants, but it involves quantum matrices and this is not a quantum matrix since it would need to satisfy $X_1 = qX_1$.