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Search: id:A110334
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| A110334 |
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Number of peakless Motzkin paths of length n having no valleys (i.e. (1,-1) followed by (1,1)) at level zero (can be easily translated into RNA secondary structure terminology). |
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+0
2
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| 1, 1, 1, 2, 4, 8, 16, 33, 70, 152, 336, 754, 1714, 3940, 9145, 21406, 50478, 119814, 286045, 686456, 1655053, 4007131, 9738812, 23750895, 58106547, 142569506, 350738607, 864980279, 2138034715, 5295877279, 13143521437, 32679745904
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Column 0 of A110333.
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REFERENCES
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W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51, 317-323, 1994.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26, 1978, 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux et problemes d'enumeration en biologie moleculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
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FORMULA
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G.f.=(3-z-z^2-Q)/(2-3z+z^2+z^3+zQ), where Q=sqrt(1-2z-z^2-2z^3+z^4).
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EXAMPLE
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a(6)=16 because among the 17 (=A004148(6)) peakless Motzkin paths of length 6 only UH(DU)HD has a valley at level 0 (shown between parentheses; here U=(1,1), H=(1,0), D=(1,-1) ).
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MAPLE
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G:=(3-z-z^2-sqrt(1-2*z-z^2-2*z^3+z^4))/(2-3*z+z^2+z^3+z*sqrt(1-2*z-z^2-2*z^3+z^4)): Gser:=series(G, z=0, 37): 1, seq(coeff(Gser, z^n), n=1..34);
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CROSSREFS
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Cf. A004148, A110333.
Sequence in context: A005821 A004149 A129986 this_sequence A084636 A088325 A006210
Adjacent sequences: A110331 A110332 A110333 this_sequence A110335 A110336 A110337
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 20 2005
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