TITLE: What is this sequence?
QUESTION [16 upvotes]: This is again a question that I asked at Stack Exchange, but got no answer so far, so I am trying here.
Let:  
$$ a_n=\sum_{k\ge0}(k+1) {n+2\brack k+2}(n+2)^kB_k$$
$B_k$ is the Bernoulli number. ${n\brack {k}}\;$ is the unsigned Stirling number of first kind,  $\left( {0\brack {0}}\;=1 \text{  and  }{{n}\brack {k}}\;=(n-1){{n-1}\brack {k}}\;+{{n-1}\brack {k-1}}\;\right)$.
From $n=0$, the first terms  are:  $ \ \  1\ ,\ 0\ ,\ -5\ ,\ 0\ ,\ 238\ ,\ 0\ ,\ -51508\ ,\ 0\ ,\ 35028576\ , ..$
The $a_n$ are all integers, and the odd-indexed $a_{2n+1}$ vanish.
A generating function should be even, but I could not find it. 
Also, any possible combinatorial interpretation (when removing the sign)?
I would welcome any help or indication on this. Thank you in advance.
EDIT(01/09/18): Actually, this is a particular case of: 
$$ a_{n,h}=\sum_{k\ge0}{k+h-1\choose k} {n\brack h+k}n^k B_k$$
Here is a table for $a_{n,h}$, for $1\le n,h \le 9$.
\begin{matrix}
n&|&a_{n,1}&a_{n,2}&a_{n,3}&a_{n,4}&a_{n,5}&a_{n,6}&a_{n,7}&a_{n,8}&a_{n,9}\\
-&&---&---&---&---&---&---&---&---&---\\
1&|&1&0&0&0&0&0&0&0&0\\
2&|&0&1&0&0&0&0&0&0&0\\
3&|&-1&0&1&0&0&0&0&0&0\\
4&|&0&-5&0&1&0&0&0&0&0\\
5&|&24&0&-15&0&1&0&0&0&0\\
6&|&0&238&0&-35&0&1&0&0&0\\
7&|&-3396&0&1281&0&-70&0&1&0&0\\
8&|&0&-51508&0&4977&0&-126&0&1&0\\
9&|&1706112&0&-408700&0&15645&0&-210&0&1\\
\end{matrix}
I know now how to show that $a_{n,h}$ is integer, and that it is zero when $n-h$ is odd. But the proofs that I have found are quite lengthy, technical, and not really enlightening about the mathematical signification of these numbers.
Any idea?

REPLY [5 votes]: I've got that $a_{n,h}$ is the coefficient of $x^{n-1}$ in
\begin{split}
&(-1)^{n+h+1} \frac{n!}{2\cdot (h-1)!}\frac{\log(1+x)^{h}\left(\coth(-\frac{n}2\log(1+x))-1\right)}{1+x} \\
=\ &(-1)^{n+h} \frac{n!}{(h-1)!}\frac{\log(1+x)^{h}}{(1-(1+x)^{-n})(1+x)}.
\end{split}
It's yet unclear if the dependency on $n$ can be eliminated to turn this expression into a generating function.