  Four-dimensional quaternion-Kaehler metrics with an isometry are determined
by solutions to the SU($\infty$) Toda equation. We set up a perturbation theory
to construct approximate solutions to the latter which can be interpreted as
membrane instanton corrections to the moduli space metric of the universal
hypermultiplet. We compute one such solution exactly up to the five-instanton
level, including all perturbative fluctuations about the instantons. The result
shows a pattern that allows us to determine the asymptotic behaviour of all
higher instanton corrections within this solution. We find that the generating
function for the leading terms of the latter is given by the Lambert
W-function.
