An algebraic summation over the set of partitions and some strange evaluations (Q1904091)

For \(\Omega\) the set of nonnegative integers, let \[ E(x|\Omega)= \sum_{k\in \tau(m, n|\Omega)} {1 \over {x_{k_1} (x_{k_1}+ x_{k_2})\cdots (x_{k_1}+ x_{k_2}+\cdots+ x_{k_n})}} \] where \(\tau(m, n|\Omega)\) is the set of all \(n\)-tuples \(\kappa= (k_1,\dots, k_n)\in \Omega^n\) with sum \(m\). Exploiting this expansion under various conditions, the author specializes to some identities of D. Knuth and B. Pittel, others involving Stirling numbers, and an extension of the generating function for Hagen-Rothe coefficients (with its Abel analogue).