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Search: id:A097613
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| A097613 |
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binom[2n-3,n-1]+ binom[2n-2,n-2]. |
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+0
4
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| 1, 2, 7, 25, 91, 336, 1254, 4719, 17875, 68068, 260338, 999362, 3848222, 14858000, 57500460, 222981435, 866262915, 3370764540, 13135064250, 51250632510, 200205672810, 782920544640
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n) = number of Dyck (2n-1)-paths with maximum pyramid size = n. A pyramid in a Dyck path is a maximal subpath of the form k upsteps immediately followed by k downsteps, and its size is k.
a(n) = total number of runs of peaks in all Dyck (n+1)-paths. A run of peaks is a maximal subpath of the form (UD)^k with k>=1. For example, a(2)=7 because the 5 Dyck 3-paths contain a total of 7 runs of peaks (in uppercase type): uuUDdd, uUDUDd, uUDdUD,UDuUDd,UDUDUD. - David Callan (callan(AT)stat.wisc.edu), Jun 07 2006
Binomial transform of A113682. - Paul Barry (pbarry(AT)wit.ie), Aug 21 2007
If Y is a fixed 2-subset of a (2n+1)-set X then a(n+1) is the number of n-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Oct 21 2007
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LINKS
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Milan Janjic, Two Enumerative Functions
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FORMULA
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G.f. (x-1)(1-1/sqrt[1-4x])/2
a(n) = ceil(A051924/2) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 16 2007
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EXAMPLE
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a(2) = 2 because UUDDUD and UDUUDD each have maximum pyramid size = 2.
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MAPLE
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Z:=(1-z-sqrt(1-4*z))/sqrt(1-4*z)/2: Zser:=series(Z, z=0, 32): seq (ceil(coeff(Zser, z, n)), n=1..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 16 2007
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CROSSREFS
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Same as A024482 except for first term.
Cf. A051924.
Adjacent sequences: A097610 A097611 A097612 this_sequence A097614 A097615 A097616
Sequence in context: A018907 A052936 A108152 this_sequence A024482 A074605 A108081
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KEYWORD
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nonn
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AUTHOR
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David Callan (callan(AT)stat.wisc.edu), Sep 20 2004
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