  A threshold graph is any graph which can be constructed from the empty graph
by repeatedly adding a new vertex that is either adjacent to every vertex or to
no vertices. The Eulerian number $\genfrac{\langle}{\rangle}{0pt}{}{n}{k}$
counts the number of permutations of size $n$ with exactly $k$ ascents.
Implicitly Beissinger and Peled proved that the number of labeled threshold
graphs on $n\ge 2$ vertices is
  \[\sum_{k=1}^{n-1}(n-k)\genfrac{\langle}{\rangle}{0pt}{}{n-1}{k-1}2^k.\]
  Their proof used generating functions. We give a direct combinatorial proof
of this result.
