On Some Finite Sums with Factorials

The summation formula $$\sum _{i=0}^{n-1}{ϵ}^{i}i!(i^k+u_k)=v_k+{ϵ}^{n-1}n!A_{k-1}(n)$$ $(\epsilon=\pm 1; k=1,2,…; u_k, v_k\in \msbm\hbox{Z}; A_{k-1}$ is a polynomial) is derived and its various aspects are considered. In particular, divisibility with respect to $n$ is investigated. Infinitely many equivalents to Kurepa’s hypothesis on the left factorial are found.