In this study, we first obtain a closed form expression for the Euler sums of skew-hyperharmonic numbers \({\widetilde{h}}_{n}^{\left( r\right) },\) which are defined as similar to the hyperharmonic numbers \(h_{n}^{\left( r\right) }\) in all respects. We then give a representation for the Euler-type sum of the numbers \(a_{n,l}^{\left( 1\right) },\) where \(a_{n,l}^{\left( r\right) }={\widetilde{h}}_{n}^{\left( r\right) }\genfrac(){0.0pt}0{n+l}{l}^{-1}\) with \({\widetilde{h}}_{n}^{\left( 1\right) }={\widetilde{H}}_{n},\) the nth skew-harmonic number. This representation enables us to show that the Euler-type sum of the numbers \(a_{n,l}^{\left( r\right) }\) can be expressed in terms of zeta values and harmonic numbers. Finally, we investigate the Dirichlet-type generating functions of the numbers \(a_{n,l}^{\left( r\right) }.\)
