  This paper is part I of a series in which we aim to show that the singular
support of the wave trace and the length spectrum of a smooth, strictly convex,
and bounded planar billiard table are generally distinct objects. We derive an
asymptotic trace formula for the regularized resolvent which is dual to the
wave trace and contains the same information. To do this, we consider a class
of periodic orbits which have nearly degenerate Poincar\'e maps and study their
leading order behavior as the deformation parameter goes to zero, generating
large coefficients in the wave trace. We also keep careful track of the Maslov
indices, which will allow us to match contributions of opposite signs in our
subsequent paper. Each cancellation of coefficients in the resolvent trace
corresponds to making the wave trace one degree smoother. The resolvent based
approach is due to Balian and Bloch and was significantly expanded upon by
Zelditch in a foundational series of papers [Zel09], [Zel04a], [Zel04c] and
[Zel00].
