A220882 Number of (n - 6)-dissections of an n-gon (equivalently, the number of three-dimensional faces of the (n-3)-dimensional associahedron) modulo the cyclic action.

1, 2, 16, 93, 505, 2548, 12400, 58140, 266550, 1198564, 5312032, 23263695, 100910001, 434217000, 1855972096, 7887862224, 33359979546, 140492933100, 589495272736, 2465455090098, 10281760786682, 42768958597992, 177499631598976, 735146520745000, 3039095720959424, 12542491305496152

Offset

6,2

Links

Table of n, a(n) for n=6..31.

Douglas Bowman and Alon Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv preprint arXiv:1209.6270 [math.CO], 2012.

Formula

See Maple code.

Maple

C:=n->binomial(2*n, n)/(n+1);
T4:=proc(n) local t1; global C;
t1 :=  (((n-3)*(n-4)^2*(n-5))/(24*n*(2*n-5)))*C(n-2);
if n mod 2 = 0 then t1:=t1+((n-4)^2/(4*n))*C(n/2-2) fi;
if n mod 3 = 0 then t1:=t1+((n-3)/9)*C(n/3-1) fi;
if n mod 6 = 0 then t1:=t1+C(n/6-1)/3 fi;
t1; end;
[seq(T4(n), n=6..40)];

Mathematica

c = CatalanNumber;
T4[n_] := Module[{t1},
t1 = (((n - 3)*(n - 4)^2*(n - 5))/(24*n*(2*n - 5)))*c[n - 2];
If[Mod[n, 2] == 0, t1 = t1 + ((n - 4)^2/(4*n))*c[n/2 - 2]];
If[Mod[n, 3] == 0, t1 = t1 + ((n - 3)/9)*c[n/3 - 1]];
If[Mod[n, 6] == 0, t1 = t1 + c[n/6 - 1]/3]; t1];
Table[T4[n], {n, 6, 40}] (* Jean-François Alcover, Dec 02 2017, from Maple *)

Crossrefs

Cf. A003444, A003445, A003450, A220881.

Sequence in context: A207558 A207413 A271273 * A295903 A362767 A141243

Adjacent sequences: A220879 A220880 A220881 * A220883 A220884 A220885

Keyword

nonn

Author

N. J. A. Sloane, Dec 28 2012

Status

approved