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

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 $P_{n}^{k}(a_{1}, a_{2}, ...)$?

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.