A098063 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).

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

Offset

0,7

Comments

Row sums yield the RNA secondary structure numbers (A004148).

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.

Links

Table of n, a(n) for n=0..76.

M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux at problemes d'enumeration en biologie moleculaire, Sem. Loth. Comb. B08l (1984) 79-86.

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.

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).

Crossrefs

Cf. A004148.

Sequence in context: A319421 A092479 A124022 * A209438 A106396 A282869

Adjacent sequences: A098060 A098061 A098062 * A098064 A098065 A098066

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 12 2004

Status

approved