A097777 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).

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

Offset

0,4

Comments

Row n contains floor(n/3) entries (n>=3).

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

T(n,0)=A098051(n).

Sum(k*T(n,k), k>=0)=A187257(n).

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..53.

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-(1-t)[zG-z/(1-z)]].

The generating function H=H(t,z) relative to the number of subwords of the form UH^bU for a fixed b>=1 satisfies H = 1+zH+z^2*H[H-1+(t-1)z^b*(H-1-zH)].

Example

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.

Maple

eq := G = 1+z*G+z^2*G*(G-1+(t-1)*(z*G-z/(1-z))): g := RootOf(eq, G): gser := simplify(series(g, z = 0, 23)): for n from 0 to 18 do P[n] := sort(coeff(gser, z, n)) end do: 1; 1; 1; for n from 3 to 18 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)-1) end do; # yields sequence in triangular form

Crossrefs

Cf. A004148, A098051, A187257

Sequence in context: A269266 A317506 A317501 * A089738 A110333 A247292

Adjacent sequences: A097774 A097775 A097776 * A097778 A097779 A097780

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 11 2004

Status

approved