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Search: id:A135336
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| A135336 |
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Number of Dyck paths of semilength n with no UUDU's starting at level 0. |
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+0
2
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| 1, 1, 2, 4, 10, 28, 85, 271, 893, 3013, 10351, 36075, 127219, 453097, 1627378, 5887660, 21436354, 78484402, 288779728, 1067263660, 3960081904, 14746806292, 55094725918, 206450572930, 775724723086, 2922060848734
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OFFSET
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0,3
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COMMENT
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Column 0 of A135330. Partial sums of the Fine sequence 1,0,1,2,6,18,... (A000957 without the first term). - 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)=Sum[(-1)^j*(3j+1)binom(2n-3j,n),j=0..floor(n/3)]/(n+1). G.f.=C/(1+z^3*C^3)=C/[(1-z)(1+zC)], 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 UUDUDD does not qualify.
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MAPLE
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a:=proc(n) options operator, arrow: (sum((-1)^j*(3*j+1)*binomial(2*n-3*j, n), j =0..floor((1/3)*n)))/(n+1) end proc: seq(a(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, A135330, A000957.
Sequence in context: A149823 A149824 A068875 this_sequence A149825 A149826 A149827
Adjacent sequences: A135333 A135334 A135335 this_sequence A135337 A135338 A135339
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), 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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