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A098073
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).
0
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
OFFSET
0,4
COMMENTS
Row sums are the RNA secondary structure numbers (A004148).
LINKS
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, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Sem. Loth. Comb. B08l (1984) 79-86.
FORMULA
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.
EXAMPLE
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).
CROSSREFS
Cf. A004148.
Sequence in context: A118424 A118429 A110317 * A118390 A247294 A202848
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 13 2004
STATUS
approved