# Positivity of q-analogues of central binomial coefficients?

With the usual $$q-$$ratings
$$[n]_q = 1 + q + cdots + q ^ {n-1} = frac {, , 1-q ^ n} {1-q},$$
$$[n]_q! =[1]_q[2]_q cdots[n]_q$$ and
$$binom {n} k_q = frac {[n]_q!} {[k]_q! cdot[n-k]_q!}$$
let
$$b (n, k, r, q) = det left (q ^ {r binom {i-j} 2} frac {[2i+k+1]_q} {[i+j+k]_q} binom {i + j + k} {i-j + 1} _q right) _ {i, j = 0} ^ {n-1}.$$

We can show that $$b (n, k, 1, q) = binom {2n + k-1} {n} _q.$$ So $$b (n, k, 1, q)$$ has positive coefficients as a polynomial in $$q$$ for each positive integer $$k$$

Calculations suggest that $$b (n, k, 0, q)$$ and $$b (n, k, 2, q) = q ^ {n (n + k-1)} b (n, k, 0,1 / q)$$ have positive coefficients.

An idea how to prove that?