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A045721
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Binomial(3n+1,n)
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13
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1, 4, 21, 120, 715, 4368, 27132, 170544, 1081575, 6906900, 44352165, 286097760, 1852482996, 12033222880, 78378960360, 511738760544, 3348108992991, 21945588357420, 144079707346575, 947309492837400, 6236646703759395
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of leaves in all noncrossing rooted trees on n nodes on a circle.
Number of standard tableaux of shape (n-1,1^(2n-3)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2004
a(n) = number of Dyck (2n-3)-paths with exactly one descent of odd length. For example, a(3) counts all 5 Dyck 3-paths except UDUDUD. - David Callan (callan(AT)stat.wisc.edu), Jul 25 2005
a(n+2) gives row sums of A119301. - Paul Barry (pbarry(AT)wit.ie), May 13 2006
a(n) is he number of paths avoiding $\uparrow ^{=2}$ from $(0,0)$ to \ $% (3n,n)$ and taking steps from \{$\uparrow ,\longrightarrow $\}. [From Shanzhen Gao (shanzhengao(AT)yahoo.com), Apr 15 2010]
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REFERENCES
| Shanzhen Gao, Pattern Avoidance in Paths and Walks, in preparation [From Shanzhen Gao (shanzhengao(AT)yahoo.com), Apr 15 2010]
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LINKS
| Milan Janjic, Two Enumerative Functions
Index entries for sequences related to rooted trees
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FORMULA
| a(n) is asymptotic to c/sqrt(n)*(27/4)^n with c=0.73... - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 27 2003
G.f.: gz^2/(1-3zg^2), where g=g(z) is given by g=1+zg^3, g(0)=1, i.e. (in Maple command) g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z); - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 22 2003
a(n+2)=C(3n+1,n)=sum{k=0..n, C(3n-k,n-k)}; - Paul Barry (pbarry(AT)wit.ie), May 13 2006
a(n+2)=C(3n+1,2n+1)=A078812(2n,n); - Paul Barry (pbarry(AT)wit.ie), Nov 09 2006
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MAPLE
| [seq( binomial(3*n+1, n), n=0..40)]; - N. J. A. Sloane (njas(AT)research.att.com), Jun 09 2007
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CROSSREFS
| Sequence in context: A020048 A093426 A046090 * A101810 A001888 A103769
Adjacent sequences: A045718 A045719 A045720 * A045722 A045723 A045724
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu)
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EXTENSIONS
| Simpler definition from Ira Gessel (gessel(AT)brandeis.edu), May 26 2007. This change means that most of the offsets in the comments will now need to be changed too.
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