OFFSET
0,4
COMMENTS
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
G.f.: 2t/(2t - 2tz - 1 + z + tz^2 + sqrt(1 - 2z - 2tz^2 + z^2 - 2tz^3 + t^2*z^4)).
EXAMPLE
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).
Triangle begins
1;
1;
1;
2;
4;
7, 1;
12, 5;
21, 15, 1;
37, 37, 8;
MAPLE
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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jul 19 2005
STATUS
approved