# What is this sequence?

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?

• I note that OP has put this sequence up on the Online Encyclopedia of Integer Sequences, oeis.org/A286483 – Gerry Myerson Jul 30 '17 at 0:56
• @Gerry, I did add it on the OEIS, because I was told it was not there, and I was hoping to advertise about it. – René Gy Jul 30 '17 at 12:53
• Might have been a good idea to mention that, so as to prevent people from wasting their time looking it up. – Gerry Myerson Jul 30 '17 at 13:07
• They seem to have alternating signs. – F. C. Jan 12 '18 at 20:08
• The diagonals $a_{n+k,n}$ for fixed even $k$ seem to be polynomials in $n$ of the form $p_k(n)(n+k+1)\cdots (n+1)n$ where $p_k(n)$ is a polynomial of degree $k-2$. – Martin Rubey Sep 1 '18 at 20:08