# Positivity of certain polynomial coefficients

Consider the rational functions (in fact, polynomials) $$F_n(q)=\frac1{(1-q)^{2n}}\sum_{k=0}^n(-q)^k\frac{2k+1}{n+k+1}\binom{2n}{n-k} \prod_{j=0,\,j\neq k}^n\frac{1+q^{2j+1}}{1+q}.$$ The numbers $\frac{2k+1}{n+k+1}\binom{2n}{n-k}$ belong to a family of Catalan triangle of which the special case $k=0$ yields the Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$.

Of further interest is $F_n(1)=E_{2n}$, the Euler numbers.

Question. Is it true that $F_n(q)$ has positive coefficients? Experiments suggest so.