OFFSET
0,4
COMMENTS
Row n (n >= 3) has floor(n/3) terms.
Row sums yield the RNA secondary structure numbers (A004148).
LINKS
W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51, 317-323, 1994.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumeration en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
FORMULA
T(n,0) = A110334(n).
Sum_{k>=0} k*T(n,k) = A110335(n-6) for n >= 6, 0 otherwise.
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).
EXAMPLE
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.
Triangle begins:
1;
1;
1;
2;
4;
8;
16, 1;
33, 4;
70, 12;
152, 32, 1;
336, 62, 5;
MAPLE
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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jul 20 2005
EXTENSIONS
Keyword tabf added by Michel Marcus, Apr 09 2013
STATUS
approved