  The present article deals with differential equations with spectral parameter
from the point of view of formal power series. The treatment does not make use
of the notion of eigenvalue, but introduces a new idea: the spectral residue.
  The article focuses on second-order, self-adjoint problems. In such a setting
every potential function determines a sequence of spectral residues. This
correspondence is invertible, and gives rise to a combinatorial inversion
formula. Other interesting combinatorial consequences are obtained by
considering spectral residues of exactly-solvable potentials of 1-dimensional
quantum mechanics.
  It is also shown that the Darboux transformation of 1-dimensional potentials
corresponds to a simple negation of the corresponding spectral residues. This
fact leads to another combinatorial inversion formula.
  Finally, there is a brief discussion of applications. The topics considered
are enumeration problems and integrable systems.
