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A000558
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Generalized Stirling numbers of second kind.
(Formerly M4213 N1758)
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4
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1, 6, 32, 175, 1012, 6230, 40819, 283944, 2090424, 16235417, 132609666, 1135846062, 10175352709, 95108406130, 925496853980, 9357279554071, 98118527430960, 1065259283215810, 11956366813630835, 138539436100687988, 1655071323662574756, 20361556640795422729
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,2
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COMMENTS
| Contribution from Olivier GERARD, Mar 25 2009: (Start)
a(n) is the number of hierarchical partitions of a set of n elements into two second level classes : k>1 subsets of [n] are further grouped in two classes.
a(n) is equivalently the number of trees of uniform height 3 with n labelled leaves, and a root of order two. (End)
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REFERENCES
| R. Fray, A generating function associated with the generalized Stirling numbers, Fib. Quart. 5 (1967), 356-366.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
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FORMULA
| E.g.f.: 1/2*(exp(exp(x)-1)-1)^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 28 2003
a(n) = sum( stirlingS2(n,k)*stirlingS2(k,2), k=0..n) [Olivier GERARD, Mar 25 2009]
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EXAMPLE
| a(2) = 1, since there is only one partition of {1,2} into two classes, and only one way to partition those classes. a(4)=32=7*1+6*3+1*7 since there are 7 ways of partitionning {1,2,3,4} into two classes (which cannot be grouped further), 6 ways of partitioning a set of 4 elements into three classes and three ways to partition three classes into two super-classes, etc. [Olivier GERARD, Mar 25 2009]
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CROSSREFS
| Cf. A000559, A046817.
Cf. A001861 for the related bicolor set partitions. [Olivier GERARD, Mar 25 2009]
Sequence in context: A137637 A125190 A180037 * A047763 A026993 A108188
Adjacent sequences: A000555 A000556 A000557 * A000559 A000560 A000561
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from David W. Wilson (davidwwilson(AT)comcast.net), Jan 13, 2000.
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