Geodesic
Generating functions for extended Stirling numbers of the first kind
Journal of integer sequences, Tome 17 (2014) no. 6
In this paper we extend the definition of Stirling numbers of the first kind by way of a special multiset. This results in a family of number triangles for which we show how to obtain ordinary generating functions for the rows and exponential generating functions for the columns. The latter are derived via a recursive process. We also indicate how to obtain formulas, in terms of factorials, generalized harmonic numbers, and polynomials, for the entries in the columns of these number triangles.
Classification : 11B73, 05A05, 05A10, 05A15, 11B37
Keywords: Stirling number of the first kind, multiset, permutation, cycle, recurrence relation, generating function 
@article{JIS_2014__17_6_a0,
     author = {Griffiths,  Martin},
     title = {Generating functions for extended {Stirling} numbers of the first kind},
     journal = {Journal of integer sequences},
     year = {2014},
     volume = {17},
     number = {6},
     zbl = {1358.11042},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2014__17_6_a0/}
}
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Griffiths,  Martin. Generating functions for extended Stirling numbers of the first kind. Journal of integer sequences, Tome 17 (2014) no. 6. http://geodesic.mathdoc.fr/item/JIS_2014__17_6_a0/