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).

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

Table of n, a(n) for n=0..52.

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

Adjacent sequences: A098070 A098071 A098072 * A098074 A098075 A098076

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 13 2004

Status

approved