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

[![Formula 1][1]][1]


[![Formula 2][2]][2]


[![Formula 3][3]][3]


[![Formula 4][4]][4]


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 $P_{n}^{k}(1, 1, 1, ...) = \binom{n-1}{k-1}$ since the definition of the Lah numbers according to Wikipedia is 

$
Lah = \frac{n!}{k!} \binom{n-1}{k-1}.
$

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


[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.