I stumbled across the following curious empirical properties of the Bernoulli polynomials $B_n(x)$. Can anyone provide a reference or proof?

Let $k\in\mathbb{Z}$, $k\geq 2$. Then (empirically):

The coefficients of $x^{n-3}$, $x^{n-5}$, $x^{n-7}, \dots$ in the polynomial $\frac{1}{n}B_n(x+k)$ are integers.

Define a sequence $a_2,a_3,\dots$ by $a_2=20$, and if $b_i=a_{i+1}-a_i$ then the sequence $b_2,b_3,\dots$ is periodic of period four, with terms $16,16,16,20,16,16,16,20,\dots$. Let $n\equiv j\,(\mathrm{mod}\,4)$, $0\leq j\leq 3$. Then the coefficient of $x^i$ in $B_n(x+k)$ is negative if and only if $n\geq a_k$ and $i=j,j+4,j+8,\dots,j+4\lfloor \frac{n-a_k}{4}\rfloor$.

I have checked this for $n\leq 324$ and $2\leq k\leq 8$.