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Search: id:A135334
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| A135334 |
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Number of Dyck paths of semilength n having no UDDU's starting at level 1. |
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+0
2
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| 1, 1, 2, 4, 10, 29, 90, 290, 960, 3246, 11164, 38934, 137358, 489341, 1757882, 6360634, 23160528, 84802606, 312041692, 1153271984, 4279311348, 15935808866, 59537435012, 223099337404, 838282693560, 3157706225584
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Column 0 of A135328. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
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REFERENCES
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A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
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FORMULA
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a(n)=2*Sum((-1)^(j-k)*(j+1)*binom(2n-2j-1,n),j=0..floor((n-1)/2))/(n+1) (n>=1). G.f.=1+zC^2/(1+z^2*C^2), where C=[1-sqrt(1-4z)]/(2z) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
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EXAMPLE
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a(3)=4 because among the 5 (=A000108(3)) Dyck paths of semilength 3 only UUDDUD does not qualify.
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MAPLE
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a:=proc(n) options operator, arrow: 2*(sum((-1)^j*(j+1)*binomial(2*n-2*j-1, n), j=0..floor((1/2)*n-1/2)))/(n+1) end proc: 1, seq(a(n), n=1..25); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
G:=1+z*C^2/(1+z^2*C^2): C:=((1-sqrt(1-4*z))*1/2)/z: Gser:=series(G, z=0, 30); seq(coeff(Gser, z, n), n=0..25); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
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CROSSREFS
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Cf. A000108, A135328.
Adjacent sequences: A135331 A135332 A135333 this_sequence A135335 A135336 A135337
Sequence in context: A148113 A005505 A148114 this_sequence A000995 A010359 A086631
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KEYWORD
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nonn
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AUTHOR
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njas, Dec 07 2007
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EXTENSIONS
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Edited and extended by Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2007
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