%I #8 Mar 06 2020 11:42:20

%S 1,1,1,2,4,7,1,12,5,21,16,37,44,1,65,113,7,114,277,32,200,655,122,1,

%T 351,1507,416,9,616,3395,1309,53,1081,7521,3877,255,1,1897,16434,

%U 10956,1074,11,3329,35502,29820,4102,79,5842,75962,78708,14532,457,1

%N Triangle read by rows: T(n,k) is number of peakless Motzkin paths of length n and having k UHH...HD's starting above level 0, where U=(1,1), H=(1,0) and D=(1,-1) (can be easily expressed using RNA secondary structure terminology).

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

%H I. L. Hofacker, P. Schuster and P. F. Stadler, <a href="https://doi.org/10.1016/S0166-218X(98)00073-0">Combinatorics of RNA secondary structures</a>, Discrete Appl. Math., 88, 1998, 207-237.

%H P. R. Stein and M. S. Waterman, <a href="https://doi.org/10.1016/0012-365X(79)90033-5">On some new sequences generalizing the Catalan and Motzkin numbers</a>, Discrete Math., 26 (1979), 261-272.

%H M. Vauchassade de Chaumont and G. Viennot, <a href="http://www.mat.univie.ac.at/~slc/opapers/s08viennot.html">Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire</a>, Sem. Loth. Comb. B08l (1984) 79-86.

%F G.f.: G=G(t, z) satisfies aG^2 + bG + c = 0, where a=z^2*(1-2z+z^2-z^3+tz-tz^2+tz^3), b=-(1-2z+2z^2-2z^3+tz^3), c=1-z.

%e Triangle starts:

%e 1;

%e 1;

%e 1;

%e 2;

%e 4;

%e 7,1;

%e 12,5;

%e 21,16;

%e 37,44,1;

%e Row n >=2 has floor((n+1)/3) terms.

%e T(6,1)=5 because we have U(UHD)DH, HU(UHD)D, U(UHD)HD, UH(UHD)D and U(UHHD)D (the pertinent subword is shown between parentheses).

%Y Cf. A004148.

%K nonn,tabf

%O 0,4

%A _Emeric Deutsch_, Sep 13 2004