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A135339
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Number of Dyck paths of semilength n having no DUDU's starting at level 1.
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3
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1, 1, 2, 4, 11, 32, 99, 318, 1051, 3550, 12200, 42520, 149930, 533890, 1917181, 6934722, 25243539, 92405718, 339940116, 1256122632, 4660081434, 17350844808, 64814186646, 242838410652, 912333763806, 3436240272972
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Column 0 of A135333. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007
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REFERENCES
| A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
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LINKS
| Guo-Niu Han, Enumeration of Standard Puzzles
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FORMULA
| G.f.=(2zC-z+C)/(1+zC), where C=[1-sqrt(1-4z)]/(2z) is the g.f. of the Catalan numbers (A000108). a(n)=binomial(2n-2,n-1)/n + Sum[(-1)^j*(j+3)binomial(2n-j-2,n),j=0..n-2]/(n+1) for n>=1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007
a(n)=A000958(n-1)+A000958(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2009]
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EXAMPLE
| a(4)=11 because among the 14 (=A000108(4)) Dyck paths of semilength 4 the following paths do not qualify: UDUDUUDD, UUDDUDUD and UDUDUDUD.
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MAPLE
| G:=(2*z*C-z+C)/(1+z*C): 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 13 2007
a:= proc (n) options operator, arrow: binomial(2*n-2, n-1)/n+(sum((-1)^j*(j+3)*binomial(2*n-j-2, n), j=0..n-2))/(n+1) end proc: 1, seq(a(n), n=1..25); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007
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CROSSREFS
| Cf. A000108, A135333.
Sequence in context: A059305 A191586 A120848 * A148170 A156043 A148171
Adjacent sequences: A135336 A135337 A135338 * A135340 A135341 A135342
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 07 2007
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EXTENSIONS
| Edited and extended by Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 13 2007
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