Recently, I was reading a blog post called *[The P-transform](https://oeis.org/wiki/User:Peter_Luschny/P-Transform#.E2.99.A6.C2.A0P-polynomials)* by Peter Luschny, where the following formulas are given:

\begin{align*}
(-1)^k\frac{n!}{k!}\mathcal P^k_n\left(1, \frac1 2, \frac2 3, \dotsc\right) & = \genfrac[]{0pt}{}n k \\
(-1)^k\frac{n!}{k!}\mathcal P^k_n(1, 1, 1, \dotsc) & = \genfrac\lvert\rvert{0pt}{}n k \\
(-1)^k\frac{n!}{k!}\mathcal P^k_n\left(1, \frac1 2, \frac1 3, \dotsc\right) & = \genfrac\{\}{0pt}{}n k \\
(-1)^k\frac{n!}{k!}\mathcal P^k_n\left(1, 0, 0, \dotsc\right) & = \delta_{n, k}.
\end{align*}


The right-hand side in the first formula is the Stirling cycle number, the right-hand side in the second formula is the Lah number, right-hand side in the third formula is Stirling set number and the right-hand side in the fourth formula is Kronecker's delta. 


It is obvious that $\mathcal P_{n}^{k}(1, 1, 1, ...) = \binom{n-1}{k-1}$ since the definition of the Lah numbers according to [Wikipedia](https://en.wikipedia.org/wiki/Lah_number) is 

$$\genfrac\lvert\rvert{0pt}{}n k = \frac{n!}{k!} \binom{n-1}{k-1}.$$

In the blog I do not find the general formula for $\mathcal P_{n}^{k}(a_{1}, a_{2}, \dotsc)$, so my question is: what is the general formula for $\mathcal P_{n}^{k}(a_{1}, a_{2}, \dotsc)$?


[1]: https://i.sstatic.net/zWIebV5n.jpg
  [2]: https://i.sstatic.net/M6aDq63p.png
  [3]: https://i.sstatic.net/M6qkP9Dp.png
  [4]: https://i.sstatic.net/XIzBp7Xc.png



Note: I have asked the same question on [Mathematics StackExchange](https://math.stackexchange.com/questions/4940765/what-is-the-formula-for-p-nk-a-1-a-2-defined-by-peter-luschn), but it didn't receive much attention, so I thought it might be beter to ask it on MathOverflow too.

**Edit:** *(some questions I came up with after posting this question)*

 - How is $P_{n}^{k} (a_1, a_2, ...)$ connected to [DeMoivre polynomials]( https://arxiv.org/abs/2203.02868) $A_{n, k}$?