  We study a recursion that generates real sequences depending on a parameter
$x$. Given a negative $x$ the growth of the sequence is very difficult to
estimate due to canceling terms. We reduce the study of the recursion to a
problem about a family of integral operators, and prove that for every
parameter value except -1, the growth of the sequence is factorial. In the
combinatorial part of the proof we show that when $x=-1$ the resulting
recurrence yields the sequence of alternating Catalan numbers, and thus has
exponential growth. We expect our methods to be useful in a variety of similar
situations.
