  Let $p,p_1,\ldots,p_m$ be positive integers with $p_1\leq p_2\leq\cdots\leq
p_m$ and $x\in [-1,1)$, define the so-called Euler type sums ${S_{{p_1}{p_2}
\cdots {p_m},p}}\left( x \right)$, which are the infinite sums whose general
term is a product of harmonic numbers of index $n$, a power of $n^{-1}$ and
variable $x^n$, by \[S_{p_1 p_2 \cdots p_m, p}(x) := \sum_{n = 1}^\infty
\frac{H_n^{(p_1)} H_n^{(p_2)} \cdots H_n^{(p_m)}} {n^p} x^n \quad (m\in
\mathbb{N} := \{1,2,3,\ldots\}), \] where $H_n^{(p)}$ is defined by the
generalized harmonic number. Extending earlier work about classical Euler sums,
we prove that whenever $p+p_1+\cdots+p_m \leq 5$, then all sums ${S_{{p_1}{p_2}
\cdots {p_m},p}}\left( 1/2\right)$ can be expressed as a rational linear
combination of products of zeta values, polylogarithms and $\log(2)$. The proof
involves finding and solving linear equations which relate the different types
of sums to each other.
