Factorial Stirling matrix and related combinatorial sequences (Q1855420)

For any integers \(n\) and \(k\) with \(0\leq k\leq n\), let \(S(n,k)\) be the Stirling numbers of the second kind. Further let \(\widetilde S_n\) be the \(n\times n\) matrix whose \((i,j)\)-entry is \(j!S(i,j)\) if \(i\geq j\), and \(0\) otherwise. The \(\widetilde S_n\) are called factorial Stirling matrices. In this paper the authors give some relationships between the Stirling matrix, the Vandermonde matrix, the Bernoulli numbers and the Euler numbers.