  We investigate expansions for connectedness functions in the random
connection model of continuum percolation in powers of the intensity.
Precisely, we study the pair-connectedness and the direct-connectedness
functions, related to each other via the Ornstein-Zernike equation. We exhibit
the fact that the coefficients of the expansions consist of sums over connected
and $2$-connected graphs. In the physics literature, this is known to be the
case more generally for percolation models based on Gibbs point processes and
stands in analogy to the formalism developed for correlation functions in
liquid-state statistical mechanics.
  We find a representation of the direct-connectedness function and bounds on
the intensity which allow us to pass to the thermodynamic limit. In some cases
(e.g., in high dimensions), the results are valid in almost the entire
subcritical regime. Moreover, we relate these expansions to the physics
literature and we show how they coincide with the expression provided by the
lace expansion.
