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A097777 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k U H^j Us for some j>0, where U=(1,1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology).

%I A097777 #10 Jan 13 2025 04:08:54
%S A097777 1,1,1,2,4,8,16,1,32,5,65,17,134,50,1,280,136,7,592,355,31,1264,904,
%T A097777 114,1,2722,2264,378,9,5906,5604,1176,49,12900,13752,3504,215,1,28344,
%U A097777 33530,10112,835,11,62608,81358,28468,2997,71,138949,196688,78576,10173,361,1
%N A097777 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k U H^j Us for some j>0, where U=(1,1) and H=(1,0) (can be easily expressed using RNA secondary structure terminology).
%C A097777 Row n contains floor(n/3) entries (n>=3).
%C A097777 Row sums are the RNA secondary structure numbers (A004148).
%C A097777 T(n,0)=A098051(n).
%C A097777 Sum(k*T(n,k), k>=0)=A187257(n).
%H A097777 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 A097777 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 A097777 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>, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08; Sem. Loth. Comb. B08l (1984) 79-86.
%F A097777 G.f. = G = G(t, z) satisfies G=1+zG+z^2*G[G-1-(1-t)[zG-z/(1-z)]].
%F A097777 The generating function H=H(t,z) relative to the number of subwords of the form UH^bU for a fixed b>=1 satisfies H = 1+zH+z^2*H[H-1+(t-1)z^b*(H-1-zH)].
%e A097777 Triangle starts:
%e A097777   1;
%e A097777   1;
%e A097777   1;
%e A097777   2;
%e A097777   4;
%e A097777   8;
%e A097777   16,1;
%e A097777   32,5;
%e A097777   65,17;
%e A097777   134,50,1;
%e A097777   280,136,7;
%e A097777   ...
%e A097777 Row n has floor(n/3) terms, n>=3.
%e A097777 T(7,1)=5 because we have H(UHU)HDD, (UHU)HHDD, (UHU)HDHD, (UHU)HDDH and (UHHU)HDD, where U=(1,1), H=(1,0) and D=(1,-1); the U H^j U's are shown between parentheses.
%p A097777 eq := G = 1+z*G+z^2*G*(G-1+(t-1)*(z*G-z/(1-z))): g := RootOf(eq, G): gser := simplify(series(g, z = 0, 23)): for n from 0 to 18 do P[n] := sort(coeff(gser, z, n)) end do: 1; 1; 1; for n from 3 to 18 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)-1) end do; # yields sequence in triangular form
%Y A097777 Cf. A004148, A098051, A187257
%K A097777 nonn,tabf
%O A097777 0,4
%A A097777 _Emeric Deutsch_, Sep 11 2004