I encountered a certain family of infinite series in some work, which is given by $$F_r(x)=\frac1{2^r}\sum_{k=0}^r\binom{r}k\frac1{1+x(2k-r)^2}.$$ I've convincing date to believe the following is true, but it needs a proof.

Question. Does this hold true? For each $r\in\mathbb{N}$, the Taylor series for $F_r(x)$ has integer coefficients.