Integral representation of certain combinatorial recurrences

Abstract

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.

A note on number triangles that are almost their own production matrix

“…The sequences a n and m n of this section actually belong to a single family of sequences that are specified in general by three parameters (two of these parameters are equal to 1 in our case). These have been studied by Martin and Kearney [5,6]. If we change the indexing of these sequences from 0 to 1, then a n corresponds to S(1, r − 3, 1) using the notation of [5], while m n corresponds to S(1, −(r + 1), 1).…”

“…These have been studied by Martin and Kearney [5,6]. If we change the indexing of these sequences from 0 to 1, then a n corresponds to S(1, r − 3, 1) using the notation of [5], while m n corresponds to S(1, −(r + 1), 1).…”