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%I #8 Mar 07 2016 11:32:46
%S 1,1,1,1,1,1,3,1,7,1,13,3,1,22,14,1,34,46,1,1,50,118,16,1,70,264,88,1,
%T 95,530,343,9,1,125,986,1066,105,1,161,1722,2857,630,2,1,203,2863,
%U 6841,2751,76,1,252,4564,15028,9746,781,1,308,7028,30778,29778,4909,30
%N Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and having k ladders.
%C A string of consecutive up steps U_1, U_2, ..., U_m and their matching down steps D_1, D_2, ..., D_m are said to form a ladder if (i) D_1, D_2, ..., D_m are consecutive steps and (ii) the sequence of pairs (U_j, D_j) (j=1,2,...,m) is maximal. For example, in the path (UU)[U]H[D]H(DD), where U=(1,1), H=(1,0), D=(1,-1), we have 2 ladders, shown between parentheses and square brackets, respectively.
%C Row sums yield the RNA secondary structure numbers (A004148). Column 1 is A002623.
%H I. L. Hofacker, P. Schuster and P. F. Stadler, <a href="http://dx.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="http://dx.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. [Formerly: Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, p. 79-86.]
%F G.f.: G=G(t, z) satisfies G=1+zG+tz^2*G(G-1)/(1-z^2+tz^2).
%e Triangle starts:
%e 1;
%e 1;
%e 1;
%e 1,1;
%e 1,3;
%e 1,7;
%e 1,13,3;
%e 1,22,14;
%e 1,34,46,1;
%e Apparently, rows 5n+1 and 5n+2 have 2n+1 terms and rows 5n+3,5n+4 and 5n+5 have 2n+2 terms.
%e T(6,2)=3 because we have (U)H(D)[U]H[D], (U)H[U]H[D](D) and (U)[U]H[D]H(D), the two ladders being shown between parentheses and square brackets, respectively.
%Y Cf. A004148, A002623.
%K nonn,tabf
%O 0,7
%A _Emeric Deutsch_, Sep 14 2004