My friend and I are trying to find a closed form for the sum
$\sum_{k\ge 0}A(n,k)k^t,$
in which the notation $A(n,k)$ is for the Eulerian number. (An alternating notation is $\left<n\atop k\right>$ if you're more familiar with the book Concrete Mathematics) It's different from the Eulerian polynomial $A_n(t)=\sum_{k\ge 0}A(n,k)t^k$, which do have a lot of studies.
We've tried using generating functions with $Li_{-t}(z)=\sum_{k\ge 0}k^tz^k$, but only to find a more complex form. Can we really have a closed form for it?