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%I #9 Jul 23 2017 02:44:34
%S 1,1,1,2,4,7,1,12,5,21,15,1,37,37,8,65,84,35,1,114,182,115,12,200,381,
%T 323,73,1,351,777,825,313,17,616,1554,1977,1087,138,1,1081,3062,4524,
%U 3291,754,23,1897,5962,9999,9063,3209,241,1,3329,11496,21515,23300
%N Triangle read by rows: T(n,k) (k>=0) is the number of RNA secondary structures of size n (i.e., with n nodes) having k arcs that are covered by other arcs.
%C Rows 0,1,2 have one term each; row n >= 3 has ceiling(n/2) - 1 terms.
%C Rows sums yield A004148.
%C T(n,0) = A005251(n+1).
%C Sum_{k>=0} k*T(n,k) = A110318(n-5).
%H W. R. Schmitt and M. S. Waterman, <a href="http://dx.doi.org/10.1016/0166-218X(92)00038-N">Linear trees and RNA secondary structure</a>, Discrete Appl. Math., 51, 317-323, 1994.
%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 (1978), 261-272.
%H M. Vauchassade de Chaumont and G. Viennot, <a href="http://www.emis.de/journals/SLC/opapers/s08viennot.html">Polynômes orthogonaux et problèmes d'énumeration en biologie moléculaire</a>, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
%F G.f.: 2t/(2t - 2tz - 1 + z + tz^2 + sqrt(1 - 2z - 2tz^2 + z^2 - 2tz^3 + t^2*z^4)).
%e T(6,1)=5 because we have 15/(24)/3/6, 16/(24)/3/5, 16/(25)/3/4, 16/2/(35)/4 and 1/26/(35)/4 (the covered arcs are shown between parentheses).
%e Triangle begins
%e 1;
%e 1;
%e 1;
%e 2;
%e 4;
%e 7, 1;
%e 12, 5;
%e 21, 15, 1;
%e 37, 37, 8;
%p G:=2*t/(2*t-2*z*t-1+z+t*z^2+sqrt(1-2*z-2*t*z^2+z^2-2*t*z^3+t^2*z^4)): Gser:=simplify(series(G,z=0,20)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 2 do print(1) od: for n from 3 to 17 do seq(coeff(t*P[n],t^k),k=1..ceil(n/2)-1) od; # yields sequence in triangular form
%Y Cf. A004148, A005252, A110318.
%K nonn,tabf
%O 0,4
%A _Emeric Deutsch_, Jul 19 2005
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