

A003449


Number of nonequivalent dissections of an ngon into n3 polygons by nonintersecting diagonals up to rotation and reflection.
(Formerly M2687)


4



1, 1, 3, 7, 24, 74, 259, 891, 3176, 11326, 40942, 148646, 543515, 1996212, 7367075, 27294355, 101501266, 378701686, 1417263770, 5318762098, 20011847548, 75473144396, 285267393358, 1080432637662, 4099856060808, 15585106611244, 59343290815356
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

4,3


COMMENTS

In other word, the number of almosttriangulations of an ngon modulo the dihedral action.
Equivalently, the number of edges of the (n3)dimensional associahedron modulo the dihedral action.
The dissection will always be composed of one quadrilateral and n4 triangles.  Andrew Howroyd, Nov 24 2017
See Theorem 30 of Bowman and Regev (although there appears to be a typo in the formula  see Maple code below).  N. J. A. Sloane, Dec 28 2012


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Andrew Howroyd, Table of n, a(n) for n = 4..200
D. Bowman and A. Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv:1209.6270, 2012.
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595601.
C. R. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370388.


MAPLE

C:=n>binomial(2*n, n)/(n+1);
T30:=proc(n) local t1; global C;
if n mod 2 = 0 then
t1:=(1/4(3/(4*n)))*C(n2) + (3/8)*C(n/21) + (13/n)*C(n/22);
if n mod 4 = 0 then t1:=t1+C(n/41)/4 fi;
else
t1:=(1/4(3/(4*n)))*C(n2) + (1/2)*C((n3)/2);
fi;
t1; end;
[seq(T30(n), n=4..40)]; # N. J. A. Sloane, Dec 28 2012


MATHEMATICA

c = CatalanNumber;
T30[n_] := Module[{t1}, If[EvenQ[n], t1 = (1/4  (3/(4*n)))*c[n  2] + (3/8)*c[n/2  1] + (1  3/n)*c[n/2  2]; If[Mod[n, 4] == 0, t1 = t1 + c[n/4  1]/4], t1 = (1/4  (3/(4*n)))*c[n2] + (1/2)*c[(n3)/2]]; t1];
Table[T30[n], {n, 4, 40}] (* JeanFrançois Alcover, Dec 14 2017, after N. J. A. Sloane *)


PROG

(PARI) \\ See A295419 for DissectionsModDihedral()
{ my(v=DissectionsModDihedral(apply(i>if(i>=3&&i<=4, y^(i3) + O(y^2)), [1..25]))); apply(p>polcoeff(p, 1), v[4..#v]) } \\ Andrew Howroyd, Nov 24 2017


CROSSREFS

A diagonal of A295634.
Cf. A003450, A295419.
Sequence in context: A148716 A148717 A148718 * A258308 A148719 A138541
Adjacent sequences: A003446 A003447 A003448 * A003450 A003451 A003452


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


EXTENSIONS

Entry revised (following Bowman and Regev) by N. J. A. Sloane, Dec 28 2012
Name clarified by Andrew Howroyd, Nov 24 2017


STATUS

approved



