Coefficients of shifted Bernoulli polynomials 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 $2\leq k\leq 8$.
 A: Let me address (1). First, you need a correction: I suppose you intended $\frac{1}{n}B_n(x+k)$ in the empirical calculations, and not $\frac{1}{n-1}B_n(x+k)$. For instance, the $x^2$ coefficient of $B_5(x+2)$ is $30$, which is a multiple of $5 = n$ but not of $4 = n-1$. 
As to the proof of the integrality of the $x^j$ coefficient of $\frac{1}{n}B_n(x+k)$ for $j \equiv n+1 \mod{2}$, it is reduced by the "umbral" relation
$$
\frac{B_n(x+1) - B_n(x)}{n} = x^{n-1} \in \mathbb{Z}[x]
$$
to the vanishing of the odd Bernoulli numbers (apart from the explicit value of $B_1 = 1/2$). Just sum this relation $k-1$ times with integer-shifted arguments. 
A: I think I see how to prove (2), or whatever version of it is true, but I haven't checked the details. Namely, use the identity
  $$ B_n(x+d) = \sum_{k=0}^n {n\choose k}B_k(d)x^{n-k}, $$
together with known results on the real zeros of Bernoulli polynomials such as 
https://ac.els-cdn.com/0021904589900166/1-s2.0-0021904589900166-main.pdf?_tid=147e6852-c89e-44e9-a805-69dd4bc54a5b&acdnat=1525395947_25f4432d47ebd7c61d3d59b2cff877a9.
