Completion of a Rational Function Sequence of Carlitz
Leonard M. Smiley
Abstract
The exponential generating functions of n(n+m) for arbitrary integer m are expressed as rational functions of the e.g.f. of n(n-1) [the tree function] and then of the e.g.f. of nn [the endofunction function]. The coefficients in these rational functions include 2nd-order Eulerian numbers (a result of L. Carlitz), 2nd-order Stirling numbers, and Stirling numbers of the first kind for negative sets (in the sense of D. Loeb). Several combinatorial identities follow.
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