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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).

%I A098071 #10 Mar 06 2020 11:42:00
%S A098071 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,
%T A098071 588,233,129,28,1376,585,237,84,1,3245,1441,468,210,9,7717,3481,1026,
%U A098071 466,45,18485,8319,2434,968,165,1,44535,19835,5972,1984,495,11,107796,47436
%N 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).
%C A098071 Row sums are the RNA secondary structure numbers (A004148).
%C A098071 T(n,0)=A190159(n).
%C A098071 Row n has 1+floor(n/3) terms.
%C A098071 Sum(k*T(n,k),k>=0) = A187260.
%H A098071 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 A098071 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 A098071 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 A098071 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.
%F A098071 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).
%e A098071 Triangle starts:
%e A098071 1;
%e A098071 1;
%e A098071 1;
%e A098071 1,1;
%e A098071 1,3;
%e A098071 2,6;
%e A098071 6,10,1;
%e A098071 17,15,5;
%e A098071 44,23,15;
%e A098071 107,42,35,1;
%e A098071 T(6,2)=1 because we have (uhd)(uhd) (the two pertinent subwords are shown between parentheses).
%Y A098071 Cf. A004148, A190159, A187260.
%K A098071 nonn,tabf
%O A098071 0,7
%A A098071 _Emeric Deutsch_, Sep 13 2004