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What is the formula for $\mathcal P_{n}^{k} (a_{1}, a_{2}, ...)$, defined by Peter Luschny?

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 …