cp's OEIS Frontend

A098056 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k subwords of the type uh^ju, dH^jd, or dh^ju for some j>0, where u=(1,1), d=(1,-1) and h=(1,0) (can be easily expressed using RNA secondary structure terminology).

%I A098056 #13 Aug 03 2025 04:54:44
%S A098056 1,1,1,2,4,8,15,2,27,9,1,48,29,5,84,80,21,147,198,74,4,257,463,230,27,
%T A098056 1,451,1033,667,125,7,796,2235,1811,488,43,1413,4727,4694,1676,219,6,
%U A098056 2526,9828,11700,5317,946,54,1,4544,20192,28252,15813,3696,326,9,8226,41100
%N A098056 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k subwords of the type uh^ju, dH^jd, or dh^ju for some j>0, where u=(1,1), d=(1,-1) and h=(1,0) (can be easily expressed using RNA secondary structure terminology).
%C A098056 Row sums are the RNA secondary structure numbers (A004148).
%C A098056 T(n,0) = A098057(n).
%C A098056 Sum(k*T(n,k),k>=0) = A187259(n).
%H A098056 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 A098056 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 A098056 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 A098056 G.f.=G=G(t, z) satisfies G = 1 + zG + z^2*[H + 2tzH/(1-z)+t^2*z^2*H/(1-z)^2+ z/(1-z)][G-(1-t)zH/(1-z)^2], where H=(1-z)^2*G-1+z.
%F A098056 The 4-variate g.f. G(t,s,v,z) of peakless Motzkin paths, where t, s, v mark subwords of the types uH^ju, dH^jd, dH^ju, respectively, and z marks length, satisfies the equation
%F A098056 G = 1+zG+z^2*[H + (t+s)zH/(1-z)+tsz^2*H/(1-z)^2+z/(1-z)][G-(1-v)zH/(1-z)^2],
%F A098056 where H = (1-z)[(1-z)G-1]. As special cases we get the current sequence A098056 and the sequences A097777 and A098083.
%e A098056 Triangle starts:
%e A098056   1;
%e A098056   1;
%e A098056   1;
%e A098056   2;
%e A098056   4;
%e A098056   8;
%e A098056   15,2;
%e A098056   27,9,1;
%e A098056   48,29,5;
%e A098056   84.80,21;
%e A098056   147,198,74,7;
%e A098056   ...
%e A098056 It seems that the number r(n) of terms in row n>=3 is given by r(n)=n/2-1 if n=2 (mod 4) and r(n)=2*round(n/4)-1 otherwise (here round(m) is the nearest integer to m).
%e A098056 T(7,1)=9 because we have h(uhu)hdd, (uhhu)hdd, (uhu)hhdd, (uhu)hddh, uh(dhu)hd and the reflections of the first four paths in a vertical axis; here u=(1,1), h=(1,0), d=(1,-1) and the pertinent subwords are shown between parentheses.
%Y A098056 Cf. A041048, A098057, A187259.
%K A098056 nonn,tabf
%O A098056 0,4
%A A098056 _Emeric Deutsch_, Sep 11 2004