login
A098071
Triangle read by rows: T(n,k) is number of peakless Motzkin paths of length n and having k uhh...hd's starting at level 0, where u=(1,1), h=(1,0) and d=(1,-1) (can be easily expressed using RNA secondary structure terminology).
2
1, 1, 1, 1, 1, 1, 3, 2, 6, 6, 10, 1, 17, 15, 5, 44, 23, 15, 107, 42, 35, 1, 252, 94, 70, 7, 588, 233, 129, 28, 1376, 585, 237, 84, 1, 3245, 1441, 468, 210, 9, 7717, 3481, 1026, 466, 45, 18485, 8319, 2434, 968, 165, 1, 44535, 19835, 5972, 1984, 495, 11, 107796, 47436
OFFSET
0,7
COMMENTS
Row sums are the RNA secondary structure numbers (A004148).
T(n,0)=A190159(n).
Row n has 1+floor(n/3) terms.
Sum(k*T(n,k),k>=0) = A187260.
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-z-z^2+2z^3-tz+2tz^2-2tz^3-tz^4+t^2z^4), b=-(1-z)(1-2z+2z^2+z^3-2tz^3), c=(1-z)^2.
The g.f. H(t,z), counting peakless Motzkin paths by the number of UH^bD (b is fixed) starting at level 0 (marked by t) and by length (marked by z), satisfies the equation H=1+zH+z^2*H(g-1-z^b + tz^b), where g=1+zg+z^2*g(g-1).
EXAMPLE
Triangle starts:
1;
1;
1;
1,1;
1,3;
2,6;
6,10,1;
17,15,5;
44,23,15;
107,42,35,1;
T(6,2)=1 because we have (uhd)(uhd) (the two pertinent subwords are shown between parentheses).
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 13 2004
STATUS
approved