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Search: id:A097777
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| A097777 |
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Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k U H^j Us for some j>0, where U=(1,1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology). |
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+0
2
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| 1, 1, 1, 2, 4, 8, 16, 1, 32, 5, 65, 17, 134, 50, 1, 280, 136, 7, 592, 355, 31, 1264, 904, 114, 1, 2722, 2264, 378, 9, 5906, 5604, 1176, 49, 12900, 13752, 3504, 215, 1, 28344, 33530, 10112, 835, 11, 62608, 81358, 28468, 2997, 71, 138949, 196688, 78576, 10173, 361, 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 G=1+zG+z^2*G[G-1-(1-t)[zG-z/(1-z)]].
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EXAMPLE
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Triangle starts:
1;
1;
1;
2;
4;
8;
16,1;
32,5;
65,17;
134,50,1;
280,136,7;
Row n has floor(n/3) terms, n>=3.
T(7,1)=5 because we have H(UHU)HDD, (UHU)HHDD, (UHU)HDHD, (UHU)HDDH and (UHHU)HDD, where U=(1,1), H=(1,0) and D=(1,-1); the U H^j U's are shown between parentheses.
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CROSSREFS
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Cf. A004148.
Sequence in context: A098864 A002546 A010745 this_sequence A089738 A110333 A069783
Adjacent sequences: A097774 A097775 A097776 this_sequence A097778 A097779 A097780
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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 11 2004
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