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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
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
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 28 2012
STATUS
approved