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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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1
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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; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Column 0 of A135309. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 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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FORMULA
| 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
a(n) = the upper left term in M^n, M = the following production matrix in which a column of (1,1,1,0,0,0,...) is prepended to an infinite lower triangular matrix with all 1's and the rest zeros:
1, 1, 0, 0, 0, 0, 0,...
1, 1, 1, 0, 0, 0, 0,...
1, 1, 1, 1, 0, 0, 0,...
0, 1, 1, 1, 1, 0, 0,...
0, 1, 1, 1, 1, 1, 0,...
0, 1, 1, 1, 1, 1, 1,...
...
- Gary W. Adamson, Jul 11 2011
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EXAMPLE
| a(4)=13 because among the 14 (=A000108(4)) Dyck paths of semilength 4 only UUUUDDDD does not qualify.
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MAPLE
| 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
| Cf. A000108, A135309.
Sequence in context: A087626 A125094 A114465 * A135337 A133365 A135335
Adjacent sequences: A135307 A135308 A135309 * A135311 A135312 A135313
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 07 2007
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2007
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