References/Proof of the conjectured identity for the Stirling permutation number $\left\{{n\atop n-k}\right\}$ While working with a combinatorics problem, I conjectured that
$$ \left\{{n \atop n-k }\right\}=\sum_{p=0}^{k-1}\bigg\langle\!\!\bigg\langle{k\atop k-1-p}\bigg\rangle\!\!\bigg\rangle \binom{n+p}{2k}, $$
where $\left\{ {n \atop k} \right\}$ is the Stirling permutation numbers and  $\big\langle\!\big\langle{n \atop k}\big\rangle\!\big\rangle $ denotes the Eulerian numbers of the second kind.


*

*All of my motivation comes from the fact that this is known to hold for $k = 1, 2, 3$. (See this, for instance.)

*I have little background on this topic, and I was unable to find this one from DLMF.

*I numerically checked that this identity holds for $n = 1, \cdots, 10$ using CAS and OEIS A008517.
Although I hardly believe that this type of identity is not known, I could not find any proof or reference to this one. So any additional information will be appreciated!

"Concrete Mathematics (what else?) - Eulerian Numbers" - says:
"Second-order Eulerian numbers are important chiefly because of their connection with Stirling numbers"
Eq. (6.43) therein gives
$$
\left\{ \matrix{  x \cr x - n \cr}  \right\}
   = \sum\limits_{\left( {0\, \le } \right)\,k\,\left( { \le \,n} \right)} {\left\langle {\left\langle \matrix{
  n \cr 
  k \cr}  \right\rangle } \right\rangle } \binom{x+n-1-k}{2n}
\quad \left| \matrix{
  \;0 \le n \in Z \hfill \cr 
  \;x \in C \hfill \cr}  \right.
$$
which easily reduce to yours, and can be extended to define Stirling No. of 2nd kind, of the indicated form, 
to complex values of $x$.
Interestingly,  also given is a twin one for the Stirling No. of 1st kind
$$
\left[ \matrix{  x \cr   x - n \cr}  \right] 
  = \sum\limits_{\left( {0\, \le } \right)\,k\,\left( { \le \,n} \right)} {\left\langle {\left\langle \matrix{
  n \cr   k \cr}  \right\rangle } \right\rangle } \binom{x+k}{2n}
  \quad \left| \matrix{
  \;0 \le n \in Z \hfill \cr 
  \;x \in C \hfill \cr}  \right.
$$
