cp's OEIS Frontend

A003445 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation.

%I A003445 M1859 #34 Jan 19 2021 11:48:00
%S A003445 1,2,8,40,165,712,2912,11976,48450,195580,784504,3139396,12526605,
%T A003445 49902440,198499200,788795924,3131945190,12428258796,49295766000,
%U A003445 195464345440,774857314042,3071175790232,12171403236288,48233597481200,191138095393700,757436171945952
%N A003445 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation.
%C A003445 In other words, the number of (n-5)-dissections of an n-gon modulo the cyclic action.
%C A003445 Equivalently, the number of two-dimensional faces of the (n-3)-dimensional associahedron modulo the cyclic action.
%C A003445 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 A003445 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003445 T. D. Noe, <a href="/A003445/b003445.txt">Table of n, a(n) for n = 5..200</a>
%H A003445 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 A003445 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 A003445 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 A003445 See Maple program.
%p A003445 C:=n->binomial(2*n,n)/(n+1);
%p A003445 T31:=proc(n) local t1; global C;
%p A003445 t1 :=  (n-3)^2*(n-4)*C(n-2)/(4*n*(2*n-5));
%p A003445 if n mod 5 = 0 then t1:=t1+(4/5)*C(n/5-1) fi;
%p A003445 if n mod 2 = 0 then t1:=t1+(n-4)*C(n/2-1)/8 fi;
%p A003445 t1; end;
%p A003445 [seq(T31(n),n=5..40)];
%t A003445 Table[t1 = (n - 3)^2*(n - 4)*CatalanNumber[n - 2]/(4*n*(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (4/5)*CatalanNumber[n/5 - 1]]; If[Mod[n, 2] == 0, t1 = t1 + (n - 4)*CatalanNumber[n/2 - 1]/8]; t1, {n, 5, 20}] (* _T. D. Noe_, Jan 03 2013 *)
%o A003445 (PARI) \\ See A295495 for DissectionsModCyclic()
%o A003445 { my(v=DissectionsModCyclic(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 A003445 A diagonal of A295633.
%Y A003445 Cf. A003444, A003450, A220881.
%Y A003445 Cf. A003443, A003448, A003450.
%K A003445 nonn
%O A003445 5,2
%A A003445 _N. J. A. Sloane_
%E A003445 Entry revised (following Bowman and Regev) by _N. J. A. Sloane_, Dec 28 2012
%E A003445 Name clarified by _Andrew Howroyd_, Nov 25 2017