A098050 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n and containing a total of k level steps H in all UHH...HD's, where U=(1,1), H=(1,0) and D=(1,-1) (can be easily expressed using RNA secondary structure terminology).

1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 5, 2, 1, 1, 16, 11, 6, 2, 1, 1, 32, 25, 14, 7, 2, 1, 1, 64, 57, 35, 17, 8, 2, 1, 1, 128, 130, 86, 46, 20, 9, 2, 1, 1, 256, 296, 212, 119, 58, 23, 10, 2, 1, 1, 512, 672, 520, 311, 156, 71, 26, 11, 2, 1, 1, 1024, 1520, 1269, 805, 428, 197, 85, 29

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 G=1+zG+z^2*G[G-1-z/(1-z)+tz/(1-tz)].

Example

Triangle starts:
1;
1;
1;
1,1;
1,2,1;
1,4,2,1;
1,8,5,2,1;
1,16,11,6,2,1;
Row n has n-1 terms, n>=2.
T(7,3)=5 because we have U(HHH)DHH, HU(HHH)DH, HHU(HHH)D, U(H)DU(HH)D,
U(HH)DU(H)D and UU(HHH)DD, where U=(1,1), H=(1,0) and D=(1,-1); the
three pertinent H's are shown between parentheses.

Crossrefs

Cf. A004148.

Sequence in context: A155038 A057728 A176463 * A278984 A111579 A144374

Adjacent sequences: A098047 A098048 A098049 * A098051 A098052 A098053

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 11 2004

Status

approved