A110333 - OEIS

%I #13 Jul 23 2017 02:44:23

%S 1,1,1,2,4,8,16,1,33,4,70,12,152,32,1,336,82,5,754,206,18,1714,512,56,

%T 1,3940,1264,163,6,9145,3109,456,25,21406,7634,1243,88,1,50478,18737,

%U 3326,284,7,119814,46006,8781,868,33,286045,113062,22955,2556,129,1,686456

%N Triangle read by rows: T(n,k) (n,k>=0) = number of peakless Motzkin paths of length n having k valleys (i.e., (1,-1) followed by (1,1)) at level zero (can be easily translated into RNA secondary structure terminology).

%C Row n (n >= 3) has floor(n/3) terms.

%C Row sums yield the RNA secondary structure numbers (A004148).

%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 T(n,0) = A110334(n).

%F Sum_{k>=0} k*T(n,k) = A110335(n-6) for n >= 6, 0 otherwise.

%F G.f.: (1 + z^2*g - tz^2*g - z^2 + tz^2)/(1 - z - z^3*g - tz^2*g + tz^3*g + z^3 + tz^2 - tz^3), 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 T(10,2)=5 because we have HUH(DU)H(DU)HD, UH(DU)H(DU)HDH, UHH(DU)H(DU)HD, UH(DU)HH(DU)HD and UH(DU)H(DU)HHD, where U=(1,1), H=(1,0), D=(1,-1) and the valleys at level zero are shown between parentheses.

%e Triangle begins:

%e     1;

%e     1;

%e     1;

%e     2;

%e     4;

%e     8;

%e    16,  1;

%e    33,  4;

%e    70, 12;

%e   152, 32,  1;

%e   336, 62,  5;

%p g:=(1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))/2/z^2: G:=(1+z^2*g-z^2*g*t-z^2+t*z^2)/(1-z-z^3*g-t*z^2*g+t*z^3*g+z^3+t*z^2-t*z^3): Gser:=simplify(series(G,z=0,23)): P[0]:=1: for n from 1 to 20 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 2 do print(1) od: for n from 3 to 20 do seq(coeff(t*P[n],t^k),k=1..floor(n/3)) od; # yields sequence in triangular form

%Y Cf. A004148, A110334, A110335.

%K nonn,tabf

%O 0,4

%A _Emeric Deutsch_, Jul 20 2005

%E Keyword tabf added by _Michel Marcus_, Apr 09 2013