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A089738
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Triangle of T(n,k)=number of peakless Motzkin paths of length n containing k valleys (can be easily expressed using RNA secondary structure terminology).
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0
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1, 1, 1, 2, 4, 8, 16, 1, 33, 4, 69, 13, 146, 38, 1, 312, 106, 5, 673, 284, 21, 1463, 742, 77, 1, 3202, 1904, 261, 6, 7050, 4823, 831, 31, 15605, 12096, 2534, 136, 1, 34705, 30106, 7474, 540, 7, 77511, 74484, 21480, 1984, 43
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Rows 0,1,2 contain one entry each and row n (n>=3) contains floor(n/3) entries.
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REFERENCES
| 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
| M. S. Waterman, Home Page (contains copies of his papers)
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
| G.f. G(t, z) satisfies G=1+zG+z^2*(G-1)[G-(1-t)(G-1-zG)].
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EXAMPLE
| T(7,1)=4 because we have HUH(DU)HD, UH(DU)HDH, UH(DU)HHD and UHH(DU)HD, where U=(1,1), D=(1,-1) and H=(1,0); the valleys are shown between parentheses.
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CROSSREFS
| Row sums give A004148.
Sequence in context: A002546 A010745 A097777 * A110333 A069783 A102251
Adjacent sequences: A089735 A089736 A089737 * A089739 A089740 A089741
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 07 2004
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