2000/06/14 by Leonard M. Smiley, Smiley, Leonard M.
Mathematics · #05A15 #05A19 #11B73 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A15 #msc:05A19 #msc:11B73
paper · pdf · doi:10.48550/arxiv.math/0006106
8 pages
arxiv created 2000/06/14 · arxiv updated 2016/09/07
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.