Let n and r be natural numbers and \(\begin{aligned} o_{n}^{\left( r\right) }=\sum _{k=1}^{n}o_{k}^{(r-1)}\text { with } o_{n}^{(0)}=\frac{1}{2n-1}, \end{aligned}\) be the hyperharmonic extension of the odd harmonic numbers \(O_{n} =1+1/3+1/5+\cdots +1/\left( 2n-1\right) \) . For this extension, we obtain a generating function, recursion relations, and closed-form evaluation formulas in terms of hyperharmonic numbers. Moreover, we show that the Euler-type sums \(\begin{aligned} \sum _{n=1}^{\infty }\frac{f_{n}}{n^{p}}\text { and }\sum \limits _{n=1}^{\infty }\frac{o_{n}^{(r)}}{(2n\pm 1)^{p}} \end{aligned}\) can be written in terms of zeta values and log-sine integrals. Here \(f_{n} \in \left\{ o_{n}^{(r)}\right. \) , \(\left. \left( -1\right) ^{n-1}{\widetilde{h}} _{n}^{\left( r\right) },\, h_{2n}^{\left( r\right) }, {\widetilde{h}}_{2n}^{\left( r\right) }\right\} \) , and \(h_{n}^{\left( r\right) }\) and \({\widetilde{h}}_{n}^{\left( r\right) }\) stand for the hyperharmonic and skew-hyperharmonic numbers, respectively. We further present evaluation formulas for the nonlinear Euler sums whose summands involve the variant of harmonic numbers, hyperharmonic number \(h_{q}^{\left( n+1\right) }\) and reciprocal binomial coefficients.
