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A003450 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation and reflection.

%I A003450 M1673 #37 Nov 29 2023 06:58:48
%S A003450 1,2,6,24,89,371,1478,6044,24302,98000,392528,1570490,6264309,
%T A003450 24954223,99253318,394409402,1565986466,6214173156,24647935156,
%U A003450 97732340680,387428854374,1535588541762,6085702368796,24116801236744,95569050564444,378718095630676
%N A003450 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation and reflection.
%C A003450 In other words, the number of (n - 5)-dissections of an n-gon modulo the dihedral action.
%C A003450 Equivalently, the number of two-dimensional faces of the (n-3)-dimensional associahedron modulo the dihedral action.
%C A003450 The dissection will always be composed of either 1 pentagon and n-5 triangles or 2 quadrilaterals and n-6 triangles. - _Andrew Howroyd_, Nov 24 2017
%D A003450 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003450 Andrew Howroyd, <a href="/A003450/b003450.txt">Table of n, a(n) for n = 5..200</a>
%H A003450 D. Bowman and A. Regev, <a href="http://arxiv.org/abs/1209.6270">Counting symmetry classes of dissections of a convex regular polygon</a>, arXiv:1209.6270 [math.CO], 2012.
%H A003450 P. Lisonek, <a href="http://dx.doi.org/10.1006/jsco.1995.1066">Closed forms for the number of polygon dissections</a>, Journal of Symbolic Computation 20 (1995), 595-601.
%H A003450 Ronald C. Read, <a href="http://dx.doi.org/10.1007/BF03031688">On general dissections of a polygon</a>, Aequat. math. 18 (1978) 370-388.
%F A003450 See Maple program.
%p A003450 C:=n->binomial(2*n,n)/(n+1);
%p A003450 T32:=proc(n) local t1; global C;
%p A003450 if n mod 2 = 0 then
%p A003450 t1 :=  (n-3)^2*(n-4)*C(n-2)/(8*n*(2*n-5));
%p A003450 if n mod 5 = 0 then t1:=t1+(2/5)*C(n/5-1) fi;
%p A003450 if n mod 2 = 0 then t1:=t1+((3*(n-4)*(n-1))/(16*(n-3)))*C(n/2-1) fi;
%p A003450 else
%p A003450 t1 :=  (n-3)^2*(n-4)*C(n-2)/(8*n*(2*n-5));
%p A003450 if n mod 5 = 0 then t1:=t1+(2/5)*C(n/5-1) fi;
%p A003450 if n mod 2 = 1 then t1:=t1+((n^2-2*n-11)/(8*(n-4)))*C((n-3)/2) fi;
%p A003450 fi;
%p A003450 t1; end;
%p A003450 [seq(T32(n),n=5..40)];
%t A003450 c = CatalanNumber;
%t A003450 T32[n_] := Module[{t1}, If[EvenQ[n], t1 = (n-3)^2*(n-4)*c[n-2]/(8*n*(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (2/5)*c[n/5-1]]; If[EvenQ[n], t1 = t1 + ((3*(n-4)*(n-1))/(16*(n-3)))*c[n/2-1]], t1 = (n-3)^2*(n-4)*c[n-2]/(8*n *(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (2/5) * c[n/5-1]]; If[OddQ[n], t1 = t1 + ((n^2 - 2*n - 11)/(8*(n-4)))*c[(n-3)/2]]]; t1];
%t A003450 Table[T32[n], {n, 5, 40}] (* _Jean-François Alcover_, Dec 11 2017, translated from Maple *)
%o A003450 (PARI) \\ See A295419 for DissectionsModDihedral()
%o A003450 { my(v=DissectionsModDihedral(apply(i->if(i>=3&&i<=5, y^(i-3) + O(y^3)), [1..30]))); apply(p->polcoeff(p, 2), v[5..#v]) } \\ _Andrew Howroyd_, Nov 24 2017
%Y A003450 A diagonal of A295634.
%Y A003450 Cf. A003449, A295419.
%Y A003450 Cf. A003444, A003445, A220881.
%K A003450 nonn
%O A003450 5,2
%A A003450 _N. J. A. Sloane_
%E A003450 Entry revised (following Bowman and Regev) by _N. J. A. Sloane_, Dec 28 2012
%E A003450 Name clarified by _Andrew Howroyd_, Nov 24 2017