OFFSET
0,7
COMMENTS
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