  This paper presents the evaluation of the Euler sums of generalized
hyperharmonic numbers $H_{n}^{\left( p,q\right) }$ \[ \zeta_{H^{\left(
p,q\right) }}\left( r\right) =\sum\limits_{n=1}^{\infty }\dfrac{H_{n}^{\left(
p,q\right) }}{n^{r}}% \] in terms of the famous Euler sums of generalized
harmonic numbers. Moreover, several infinite series, whose terms consist of
certain harmonic numbers and reciprocal binomial coefficients, are evaluated in
terms of Riemann zeta values.
