Many recurrences that occur in combinatorics incorporate linear and self-convolutive terms. The generating function associated to these is usually not well defined because it has zero radius of convergence. However, the sequence may be identifiable as the asymptotic expansion of a function, and then contour integration can be applied to obtain an expression as the moment sequence of a (possibly signed) measure. We find examples that in combinatorics are all connected with permutations, and whose generating functions are related to the exponential integral function.
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