### A divertent generating function that can be summed and analysed analytically

*Svante Janson*

#### Abstract

We study a recurrence relation, where the generating function,
as a formal power series, satisfies a differential equation that can be
solved in a suitable domain; this yields an analytic function in a domain,
but the solution is singular at the origin and the generating function has
radius of convergence 0. Nevertheless, the solution to the recurrence can
be obtained from the analytic solution by finding an asymptotic series
expansion. Conversely, the analytic solution can be obtained by summing
the generating function by the Borel summation method

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