Recently, I was reading a blog post called The P-transform 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 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 $\mathcal P_{n}^{k}(a_{1}, a_{2}, \dotsc)$?

Note: I have asked the same question on Mathematics StackExchange, but it didn't receive much attention, so I thought it might be beter to ask it on MathOverflow too.