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A098063
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Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n and having k level steps at height >0 (can be easily expressed using RNA secondary structure terminology).
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0
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1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 7, 2, 1, 1, 9, 13, 11, 2, 1, 1, 12, 28, 22, 16, 2, 1, 1, 16, 46, 64, 33, 22, 2, 1, 1, 20, 80, 118, 126, 46, 29, 2, 1, 1, 25, 120, 258, 248, 225, 61, 37, 2, 1, 1, 30, 185, 438, 668, 460, 374, 78, 46, 2, 1, 1, 36, 260, 813, 1231, 1506, 782, 588, 97
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| Row sums yield the RNA secondary structure numbers (A004148).
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REFERENCES
| 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
| 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=G(t, z) satisfies z(t-tz+tz^2-1+2z-z^2)G^2-(1-2z+z^2+tz)G+1=0.
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EXAMPLE
| Triangle starts:
1;
1;
1;
1,1;
1,2,1;
1,4,2,1;
1,6,7,2,1;
1,9,13,11,2,1;
Row n (n>=2) has n-1 terms.
T(5,2)=2 because among the eight peakless Motzkin paths of length 5 only HU(HH)D and U(HH)DH have two H's at positive height (shown between parentheses); here U=(1,1), H=(1,0), D=(1,-1).
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CROSSREFS
| Cf. A004148.
Sequence in context: A089688 A092479 A124022 * A106396 A140998 A048004
Adjacent sequences: A098060 A098061 A098062 * A098064 A098065 A098066
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 12 2004
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