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Search: id:A094507
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| A094507 |
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Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having k UDUD's (here U=(1,1), D=(1,-1)). |
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+0
1
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| 1, 1, 1, 1, 3, 1, 1, 7, 5, 1, 1, 19, 14, 7, 1, 1, 53, 46, 22, 9, 1, 1, 153, 150, 82, 31, 11, 1, 1, 453, 495, 299, 127, 41, 13, 1, 1, 1367, 1651, 1087, 507, 181, 52, 15, 1, 1, 4191, 5539, 3967, 1991, 781, 244, 64, 17, 1, 1, 13015, 18692, 14442, 7824, 3271, 1128, 316, 77, 19
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OFFSET
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0,5
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COMMENT
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Column k=0 is A078481. Row sums are the Catalan numbers (A000108).
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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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G.f.=G=G(t, z) satisfies the equation z(1+z-tz)G^2-(1+z+z^2-tz-tz^2)G+1+z-tz=0.
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EXAMPLE
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T(3,0)=3 because UDUUDD, UUDDUD, and UUUDDD are the only Dyck paths of semilength 3 and having no UDUD in them.
Triangle begins:
[1];
[1];
[1,1];
[3,1,1];
[7,5,1,1];
[19,14,7,1,1];
[53,46,22,9,1,1];
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CROSSREFS
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Cf. A078481, A000108.
Sequence in context: A108625 A118801 A080936 this_sequence A065625 A130749 A008277
Adjacent sequences: A094504 A094505 A094506 this_sequence A094508 A094509 A094510
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 05 2004
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