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%I #4 Mar 30 2012 17:35:59
%S 1,1,1,1,1,1,2,1,1,4,2,1,1,8,5,2,1,1,16,11,6,2,1,1,32,25,14,7,2,1,1,
%T 64,57,35,17,8,2,1,1,128,130,86,46,20,9,2,1,1,256,296,212,119,58,23,
%U 10,2,1,1,512,672,520,311,156,71,26,11,2,1,1,1024,1520,1269,805,428,197,85,29
%N Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n and containing a total of k level steps H in all UHH...HD's, where U=(1,1), H=(1,0) and D=(1,-1) (can be easily expressed using RNA secondary structure terminology).
%C Row sums yield the RNA secondary structure numbers (A004148).
%D I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
%D P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26, 1979, 261-272.
%D 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.
%H M. Vauchassade de Chaumont and G. Viennot, <a href="http://www.mat.univie.ac.at/~slc/opapers/s08viennot.html">Polynomes orthogonaux at problemes d'enumeration en biologie moleculaire</a>, Sem. Loth. Comb. B08l (1984) 79-86.
%F G.f.=G=G(t, z) satisfies G=1+zG+z^2*G[G-1-z/(1-z)+tz/(1-tz)].
%e Triangle starts:
%e 1;
%e 1;
%e 1;
%e 1,1;
%e 1,2,1;
%e 1,4,2,1;
%e 1,8,5,2,1;
%e 1,16,11,6,2,1;
%e Row n has n-1 terms, n>=2.
%e T(7,3)=5 because we have U(HHH)DHH, HU(HHH)DH, HHU(HHH)D, U(H)DU(HH)D,
%e U(HH)DU(H)D and UU(HHH)DD, where U=(1,1), H=(1,0) and D=(1,-1); the
%e three pertinent H's are shown between parentheses.
%Y Cf. A004148.
%K nonn,tabf
%O 0,7
%A _Emeric Deutsch_, Sep 11 2004
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