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Broder Andrei Z., <a href="httphttps://infolab.stanforddoi.eduorg/TR10.1016/CS-TR0012-82365X(84)90161-949.html4">The r-Stirling numbers</a>, Discrete Math. 49, 241-259 (1984)

Erich Neuwirth, <a href="httphttps://homepage.univie.acdoi.atorg/erich10.neuwirth/papers1016/TechRep99S0012-365X(00)00373-05.pdf3">Recursively defined combinatorial functions: Extending Galton's board</a>, Discrete Math. 239 No. 1-3, 33-51 (2001).

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See A049458 for a signed version of this array. The unsigned 3-Stirling numbers of the first kind count the number of permutations of the set {1,2,...,n} into k disjoint cycles, with the restriction that the elements 1, 2 and 3 belong to distinct cycles. This is the case r = 3 of the unsigned r-Stirling numbers of the first kind. For other cases see abs(A008275) (r = 1), A143491 (r = 2) and A143493 (r = 4). See A143495 for the corresponding 3-Stirling numbers of the second kind. The theory of r-Stirling numbers of both kinds is developed in [Broder]. For details of the related 3-Lah numbers see A143498.

With offset n=0 and k=0, this is the Sheffer triangle (1/(1-x)^3, -log(1-x)) (in the umbral notation of S. Roman's book this would be called Sheffer for (exp(-3*t), 1-exp(-t))). See the e.g.f given below. Compare also with the e.g.f. for the signed version A049458. - Wolfdieter Lang, Oct 10 2011

T(n,k) = (n-3)! * Sum_{j = k-3 .. n-3} C(n-j-1,2)*|stirling1Stirling1(j,k-3)|/j!.

If we define f(n,i,a) = sum(binomial(n,k)*stirling1Stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then T(n,i) = |f(n,i,3)|, for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

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Erich Neuwirth Erich, , <a href="http://homepage.univie.ac.at/erich.neuwirth/papers/TechRep99-05.pdf">Recursively defined combinatorial functions: Extending Galton's board</a>, Discrete Math. 239 No. 1-3, 33-51 (2001).

Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.

Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31

Neuwirth Erich, Askar Dzhumadil’daev and Damir Yeliussizov, <a href="http://homepage.univiewww.accombinatorics.atorg/ojs/erichindex.neuwirthphp/eljc/article/papersview/TechRep99-05.pdfv22i4p10">Recursively defined combinatorial functions: Extending Galton's boardWalks, partitions, and normal ordering</a>, Discrete Math. 239 No. 1-3, 33-51 Electronic Journal of Combinatorics, 22(4) (20012015), #P4.10.

Neuwirth Erich, <a href="http://homepage.univie.ac.at/erich.neuwirth/papers/TechRep99-05.pdf">Recursively defined combinatorial functions: Extending Galton's board</a>, Discrete Math. 239 No. 1-3, 33-51 (2001).

Michael J. Schlosser and Meesue Yoo, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p31">Elliptic Rook and File Numbers</a>, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.

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