OFFSET
0,6
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1274
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.
FORMULA
G.f.: G(t, z)=(g+zg-tz-2zg^2+t^2*(1-t)z^3*g^2-2t(1-t)z^2*g)/(1-tzg)^2, where g=1+zg^3 is the g.f. for the ternary numbers (A001764).
EXAMPLE
T(2,0)=T(2,1)=0, T(2,2)=3 because in all the noncrossing trees _\, /\ and /_, the maximal number of contiguous border edges starting from the root in both directions is equal to 2.
Triangle starts:
1;
0, 1;
0, 0, 3;
1, 4, 3, 4;
7, 20, 15, 8, 5;
42, 102, 72, 36, 15, 6;
...
MAPLE
G:=(g+z*g-t*z-2*z*g^2+t^2*(1-t)*z^3*g^2-2*t*(1-t)*z^2*g)/(1-t*z*g)^2: z:=w^2: b:=w*sqrt(3): g:=2*sin(arcsin(3*b/2)/3)/b: Gser:=simplify(series(G, w=0, 24)): P[0]:=1: for n from 1 to 10 do P[n]:=sort(coeff(Gser, w^(2*n))) od: for n from 0 to 10 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form
MATHEMATICA
max = 20; z = w^2; b = w*Sqrt[3]; g = 2*(Sin[ ArcSin[3*(b/2)]/3]/b); gf = (g + z*g - t*z - 2*z*g^2 + t^2*(1 - t)*z^3*g^2 - 2*t*(1 - t)*z^2*g)/(1 - t*z*g)^2; se = Series[gf, {w, 0, max}]; Flatten[ Rest /@ DeleteCases[ (CoefficientList[t*#1, t] & ) /@ CoefficientList[se, w], {}]] (* Jean-François Alcover, Oct 05 2011, after Maple *)
PROG
(PARI)
S(n)={my(g=1+serreverse(x/(1+x)^3 + O(x*x^n))); Vec((g + x*g - y*x - 2*x*g^2 + y^2*(1-y)*x^3*g^2 - 2*y*(1-y)*x^2*g)/(1 - y*x*g)^2)}
my(v=S(10)); for(n=1, #v, my(p=v[n]); for(k=0, n-1, print1(polcoeff(p, k), ", ")); print); \\ Andrew Howroyd, Nov 17 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jan 22 2005
STATUS
approved