We invoke the following lemma:
Lemma. Let $\mathcal{F}$ be a family of formal power series about $0$ defined as follows:
$$ \mathcal{F} = \left\{ \sum_{k=0}^{\infty} \frac{a_k}{k!} x^k \,\middle|\, a_k \in \mathbb{Z} \text{ for all } k \right\} $$
Then for any $f(x), g(x) \in \mathcal{F}$, the following properties hold:
- $f(x) + g(x) \in \mathcal{F}$.
- $f(x)g(x) \in \mathcal{F}$.
- $f(g(x)) \in \mathcal{F}$, provided $g(0) = 0$.
To make use of this lemma, we substitute $x = (t + 1)/e$ and note that
\begin{align*}
A_n
&= \left[ \left(\frac{\mathrm{d}}{\mathrm{d}x}\right)^n \exp\left(\frac{n}{\log x} \right) \right]_{x=1/e} \\
&= \left[ \left(e \frac{\mathrm{d}}{\mathrm{d}t}\right)^n \exp\left(- \frac{n}{1 - \log (1+t)} \right) \right]_{t = 0} \\
&= \left[ \left(\frac{\mathrm{d}}{\mathrm{d}t}\right)^n \exp\left(n - \frac{n}{1 - \log (1+t)} \right) \right]_{t = 0}.
\end{align*}
Now we note that both
$$ \log(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} x^n
\qquad\text{and}\qquad
\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n $$
are members of $\mathcal{F}$. Moreover, $\log(1+0) = 0$. So by the lemma, we get
$$ \frac{1}{1-\log(1+x)} \in \mathcal{F}. $$
Using this and $\exp(x) \in \mathcal{F}$ together, it follows that $A_n \in \mathbb{Z}$ as desired.
For $B_n$'s, note that
\begin{align*}
B_n
&= \left[ \left(\frac{\mathrm{d}}{\mathrm{d}x}\right)^n \exp\left(\frac{n}{\log x} \right) \right]_{x=e} \\
&= \left[ \left(e^{-1} \frac{\mathrm{d}}{\mathrm{d}t}\right)^n \exp\left(\frac{n}{\log (1+t) + 1} \right) \right]_{t = 0} \tag{$x=(1+t)e$} \\
&= \left[ \left(\frac{\mathrm{d}}{\mathrm{d}t}\right)^n \exp\left(-n + \frac{n}{1 + \log (1+t)} \right) \right]_{t = 0}
\end{align*}
and a similar argument applies.
Proof of Lemma. Item 1 and 2 are straightforward, so we only focus on establishing Item 3. Since $g(0) = 0$, $h(x) = f(g(x))$ is a well-defined formal power series. Moreover, by the combinatorial form of Faà di Bruno's formula,
\begin{align*}
h^{(n)}(0)
= \sum_{\pi\in \Pi}f^{(|\pi|)}(0) \prod_{B \in \pi} g^{(|B|)}(0)
\end{align*}
where the $\pi$ runs through the set $\Pi$ of all partitions of $\{ 1, 2, \ldots, n\} $, and $B \in \pi$ means the variable $B$ runs through the list of all of the "blocks" of the partition $\pi$. The hypothesis of the lemma tells that $f^{(j)}(0)$'s and $g^{(j)}(0)$'s are all integers, hence $h^{(n)}(0)$ is also an integer and the proof is done.
