Let $a(n,k)=(-1)^k {{2n-k}\choose k}$ for $0 \le k \le n$ and $a(n,k)=0$ else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of $(a(i,j))_{i,j\ge0}$ is the sequence $1,1,2,8,56,608,9440,…$ of median Genocchi numbers.
Now let $b(n,k)=(-1)^k {{2n+1-k}\choose k}$ for $0 \le k \le n$ and $b(n,k)=0$ else.
My question is: Is a formula known for the entries of the first column of the inverse matrix of $(b(i,j))_{i,j\ge0}?$ The first terms of this sequence are $1, 1/2, 1/3, 1/3, 8/15, 4/3, 512/105, 368/15…$. Multiplying the $n-$th term with $n!$ gives the sequence $1,1,2,8,64,960,24576,989184,…$.
Edit. Motivated by the comments below and analogous results by D. Dumont and J. Zeng about Genocchi numbers I found the following connections with Bernoulli numbers $B_{2n}.$
1) Let $\left( {c(n)} \right)_{n \ge 0}=\left({1, 1/2, 1/3, 1/3, 8/15, 4/3, 512/105, 368/15…}\right).$
Then there is an expansion into a formal power series
$\sum\limits_{n \ge 0} {( - 1)^n c(n)\frac{{z^{2n} }}{{(1 - z)^{n + 1} }}} = z + \sum\limits_{n \ge 0} {(2n + 1)B_{2n} z^{2n} } .$
2) $c(2n + 1) = \sum\limits_{j = 0}^n {{2n+1}\choose{2j+1}
} (2n + 2j + 3)B_{2n + 2j + 2} $ and $c(2n ) = -\sum\limits_{j = 0}^{n-1} {{2n}\choose{2j+1}
} (2n + 2j + 3)B_{2n + 2j + 2}. $
There remains the question whether the sequence $(c(n))$ also occurs in a natural way in other contexts.
