  Recently, Dil and Boyadzhiev \cite{AD2015} proved an explicit formula for the
sum of multiple harmonic numbers whose indices are the sequence $\left(
{{{\left\{ 0 \right\}}_r},1} \right)$. In this paper we show that the sums of
multiple harmonic numbers whose indices are the sequence $\left( {{{\left\{ 0
\right\}}_r,1};{{\left\{ 1 \right\}}_{k-1}}} \right)$ can be expressed in terms
of (multiple) zeta values, multiple harmonic numbers and Stirling numbers of
the first kind, and give an explicit formula.
