  The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical
harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers:
  {\sigma}(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in
terms of series of Hurwitz zeta function values. This is a generalization of a
result of Mez\H{o} and Dil. We also provide an explicit evaluation of
{\sigma}(r,m) in a closed form in terms of zeta values and Stirling numbers of
the first kind. Furthermore, we evaluate several other series involving
hyperharmonic numbers.
