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Search: id:A132081
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| A132081 |
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Triangle (read by rows) with row sums = Motzkin numbers (A005043): T(n,s) = (1/n) C(n,s)[ C(n-s,s+1) - C(n-s-2,s-1). |
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1
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| 1, 1, 2, 1, 5, 1, 9, 5, 1, 14, 21, 1, 20, 56, 14, 1, 27, 120, 84, 1, 35, 225, 300, 42, 1, 44, 385, 825, 330, 1, 54, 616, 1925, 1485, 132, 1, 65, 936, 4004, 5005, 1287, 1, 77, 1365, 7644, 14014, 7007, 429
(list; graph; listen)
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OFFSET
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3,3
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COMMENT
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Whereas A005043 counts certain trees, or noncrossed partitions, this subdivides the counts according to the number of leaves, or the lattice rank. Analogous to the Narayana triangle (A001263), where rows sum to the Catalan numbers.
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REFERENCES
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F. R. Bernhart, Catalan, Motzkin, and Riordan numbers, Disc. Math., 204 (1999) 73- (as given in A005043)
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FORMULA
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a(n,k) = Binomial[n,k]Binomial[n-2-k,k]/(k+1). - David Callan (callan(AT)stat.wisc.edu), Jul 22 2008
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EXAMPLE
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A005043(6) = 15 = 1+9+5 since NC (noncrossed, planar) partitions of 6-point cycle without singletons have 1,9,5 items with 1,2,3 blocks.
Triangle begins:
1,
1,2,
1,5,
1,9,5,
1,14,21,
1,20,56,14,
1,27,120,84,
1,35,225,300,42,
1,44,385,825,330, ...
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CROSSREFS
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Adjacent sequences: A132078 A132079 A132080 this_sequence A132082 A132083 A132084
Sequence in context: A064865 A093127 A115123 this_sequence A054251 A119763 A092142
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KEYWORD
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nonn,tabf
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AUTHOR
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Frank R. Bernhart (farb45(AT)gmail.com), Oct 30 2007
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EXTENSIONS
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Edited by njas, Jul 01 2008 at the suggestion of R. J. Mathar
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