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Search: id:A135310
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| A135310 |
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Number of Dyck paths of semilength n having no UUUU's starting at level 0. |
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+0
2
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| 1, 1, 2, 5, 13, 36, 105, 319, 1002, 3235, 10685, 35970, 123045, 426667, 1496782, 5303623, 18956417, 68270576, 247518777, 902708185, 3309559838, 12190954231, 45096739797, 167462013888, 624019924009, 2332697899665
(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 A135309. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 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)=Sum[(-1)^(j)*(5j+1)*binom(2n-3j,n+j)/(n+j+1),j=0..floor(n/4)]. G.f.=C/[1+z^4*C^5], where C=[1-sqrt(1-4z)]/(2z) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2007
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EXAMPLE
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a(4)=13 because among the 14 (=A000108(4)) Dyck paths of semilength 4 only UUUUDDDD does not qualify.
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MAPLE
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a:=proc(n) options operator, arrow: sum((-1)^j*(5*j+1)*binomial(2*n-3*j, n+j)/(n+j+1), j=0..floor((1/4)*n)) end proc: seq(a(n), n=0..25); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2007
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CROSSREFS
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Cf. A000108, A135309.
Adjacent sequences: A135307 A135308 A135309 this_sequence A135311 A135312 A135313
Sequence in context: A087626 A125094 A114465 this_sequence A135337 A133365 A135335
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KEYWORD
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nonn,easy
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AUTHOR
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njas, Dec 07 2007
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EXTENSIONS
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More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2007
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