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Search: id:A098073
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| A098073 |
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Triangle read by rows: T(n,k) is number of peakless Motzkin paths of length n and having k UHH...HD's starting above level 0, where U=(1,1), H=(1,0) and D=(1,-1) (can be easily expressed using RNA secondary structure terminology). |
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+0
1
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| 1, 1, 1, 2, 4, 7, 1, 12, 5, 21, 16, 37, 44, 1, 65, 113, 7, 114, 277, 32, 200, 655, 122, 1, 351, 1507, 416, 9, 616, 3395, 1309, 53, 1081, 7521, 3877, 255, 1, 1897, 16434, 10956, 1074, 11, 3329, 35502, 29820, 4102, 79, 5842, 75962, 78708, 14532, 457, 1
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Row sums are the RNA secondary structure numbers (A004148).
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REFERENCES
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I. L. Hofacker, P. Schuster, and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26, 1979, 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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LINKS
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M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux at problemes d'enumeration en biologie moleculaire, Sem. Loth. Comb. B08l (1984) 79-86.
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FORMULA
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G.f.=G=G(t, z) satisfies aG^2 + bG + c = 0, where a=z^2*(1-2z+z^2-z^3+tz-tz^2+tz^3), b=-(1-2z+2z^2-2z^3+tz^3), c=1-z.
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EXAMPLE
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Triangle starts:
1;
1;
1;
2;
4;
7,1;
12,5;
21,16;
37,44,1;
Row n >=2 has floor((n+1)/3) terms.
T(6,1)=5 because we have U(UHD)DH, HU(UHD)D, U(UHD)HD, UH(UHD)D, and U(UHHD)D (the pertinent subword is shown between parentheses).
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CROSSREFS
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Cf. A004148.
Sequence in context: A118424 A118429 A110317 this_sequence A118390 A134974 A133292
Adjacent sequences: A098070 A098071 A098072 this_sequence A098074 A098075 A098076
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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), Sep 13 2004
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