%I #17 Mar 06 2020 11:42:35
%S 1,1,1,1,1,1,3,1,6,1,1,11,5,1,20,15,1,1,36,38,7,1,64,91,28,1,1,113,
%T 211,89,9,1,199,477,255,45,1,1,350,1059,690,172,11,1,615,2321,1797,
%U 572,66,1,1,1080,5037,4555,1754,295,13,1,1896,10847,11320,5098,1118,91,1,1,3328
%N Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and height k (can be easily expressed using RNA secondary structure terminology).
%C Row sums are the RNA secondary structure numbers (A004148). Column 1 without the zeros yields A077855.
%H Alois P. Heinz, <a href="/A098076/b098076.txt">Rows n = 0..200, flattened</a>
%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.: 1+sum(t^j*z^(2j+1)/[P(j)*P(j+1)], j=0..infinity), where P(j) are polynomials in z defined by P(0)=1, P(1)=1-z, P(j)=(1-z+z^2)P(j-1) -z^2*P(j-2), j=2, 3, ... .
%e Triangle starts:
%e 1;
%e 1;
%e 1;
%e 1, 1;
%e 1, 3;
%e 1, 6, 1;
%e 1, 11, 5;
%e 1, 20, 15, 1;
%e 1, 36, 38, 7;
%e 1, 64, 91, 28, 1;
%e 1, 113, 211, 89, 9;
%e 1, 199, 477, 255, 45, 1;
%e Row n >0 has ceil(n/2) terms.
%e T(6,2) = 5 because the peakless Motzkin paths of length 6 and height 2 are HUUHDD, UHUHDD, UUHHDD, UUHDDH, UUHDHD, where U=(1,1), H=(1,0) and D=(1,-1).
%p P[0]:=1: P[1]:=sort(1-z): for j from 2 to 30 do P[j]:=sort(expand((1-z+z^2)*P[j-1]-z^2*P[j-2])) od: G:=1+sum(t^i*z^(2*i+1)/P[i]/P[i+1],i=0..25): Gser:=simplify(series(G,z=0,21)): Q[0]:=1: for m from 1 to 18 do Q[m]:=sort(coeff(Gser,z^m)) od: 1,seq(seq(coeff(t*Q[n],t^k),k=1..ceil(n/2)),n=1..16);
%t max = 16; p[0] = 1; p[1] := 1 - z; p[j_] := p[j] = (1 - z + z^2)*p[j - 1] - z^2*p[j - 2]; gf = 1 + Sum[t^j*z^(2*j + 1)/(p[j]*p[j + 1]), {j, 0, max}]; se = Series[gf, {t, 0, max}, {z, 0, max}]; CoefficientList[se, {z, t}] // DeleteCases[#, 0, 2] & // Flatten (* _Jean-François Alcover_, Jun 25 2013 *)
%Y Cf. A004148, A077855.
%K nonn,tabf
%O 0,7
%A _Emeric Deutsch_, Sep 13 2004