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.



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> **Question.** Does this hold true? For each $r\in\mathbb{N}$, the Taylor series for $F\_r(x)$ has integer coefficients.
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Prove a family of series having integer coefficients
Yes. The coefficient of $x^n$ in $F\_r$ is the $2n$-th derivative at $t = 0$ of the function
$$
t \mapsto (\cos t)^{r} = \frac{1}{2^r} \sum\_{k=0}^r \binom{r}{k} e^{it(2k-r)}.
$$
But the successive derivatives of $(t \mapsto (\cos t)^r)$ are easily seen by induction to be polynomials with integer coefficients in $\cos(t),\sin(t)$.


