%I A110237
%S A110237 1,2,3,1,6,4,13,10,1,28,24,6,62,59,21,1,140,144,62,8,320,350,174,36,1,
%T A110237 740,852,474,128,10,1728,2077,1263,410,55,1,4068,5072,3318,1240,230,12,
%U A110237 9645,12412,8634,3608,835,78,1,23010,30440,22314,10216,2792,376,14
%N A110237 Triangle read by rows: T(n,k) (0<=k<=ceil(n/2)-1) is the number of (1,
               0) steps at level k in all peakless Motzkin paths of length n (can 
               be easily translated into RNA secondary structure terminology).
%C A110237 Row n has ceil(n/2) terms. Row sums yield A110236.
%D A110237 W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, 
               Discrete Appl. Math., 51, 317-323, 1994.
%D A110237 P. R. Stein and M. S. Waterman, On some new sequences generalizing the 
               Catalan and Motzkin numbers, Discrete Math., 26, 1979, 261-272.
%D A110237 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.
%F A110237 G.f.=z*g^2/(1-tz^2*g^2), where g=1+zg+z^2*g(g-1)=[1-z+z^2-sqrt(1-2z-z^2-2z^3+z^4)]/
               (2z^2) is the g.f. of the RNA secondary structure numbers (A004148).
%e A110237 T(5,1)=10 because in the 8 (=A004148(5)) peakless Motzkin paths of length 
               5, namely HHHHH, U(H)DHH, U(HH)DH, U(HHH)D, HU(H)DH, HU(HH)D, HHU(H)D 
               and UUHDD (where U=(1,1), H=(1,0) and D=(1,-1)), we have alltogether 
               10 H steps at level 1 (shown between parentheses).
%e A110237 Triangle starts:
%e A110237 1;
%e A110237 2;
%e A110237 3,1;
%e A110237 6,4;
%e A110237 13,10,1;
%p A110237 g:=(1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))/2/z^2: G:=z*g^2/(1-t*z^2*g^2): 
               Gser:=simplify(series(G,z=0,20)): for n from 1 to 15 do P[n]:=coeff(Gser,
               z^n) od: for n from 1 to 15 do seq(coeff(t*P[n],t^k),k=1..ceil(n/
               2)) od;
%Y A110237 Cf. A004148, A110236.
%Y A110237 Sequence in context: A016730 A114576 A116468 this_sequence A076631 A035485 
               A074306
%Y A110237 Adjacent sequences: A110234 A110235 A110236 this_sequence A110238 A110239 
               A110240
%K A110237 nonn,tabf
%O A110237 1,2
%A A110237 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 17 2005