The fact that $-1 <= \pi(j) - j$ means that an element of the permutation can
only shift one position to the right. This means that a permutation satisfying
the lower bound is composed of nonoverlapping factors of the form
$i+m,i,i+1,\dots,i+m-1$ that start at position $i$. The upper bound restricts the
size of $m \le k$. Since the size of the factor is $m+1$ the number of permutations
in $\mathfrak{S}_n$
satisfying $-1 \le \pi(j) - j \le k$ is just the number of compositions of $n$
whose largest part is less than or equal to $k+1$.

I don't know of a nice closed form, but
$$\#\mathcal{A}_n^{(k)}=[x^n]\frac{1-x}{1-2x+x^{k+2}}$$