  We consider a self-convolutive recurrence whose solution is the sequence of
coefficients in the asymptotic expansion of the logarithmic derivative of the
confluent hypergeometic function $U(a,b,z)$. By application of the Hilbert
transform we convert this expression into an explicit, non-recursive solution
in which the $n$th coefficient is expressed as the $(n-1)$th moment of a
measure, and also as the trace of the $(n-1)$th iterate of a linear operator.
Applications of these sequences, and hence of the explicit solution provided,
are found in quantum field theory as the number of Feynman diagrams of a
certain type and order, in Brownian motion theory, and in combinatorics.
