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 A003445 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation. (Formerly M1859) 6
 1, 2, 8, 40, 165, 712, 2912, 11976, 48450, 195580, 784504, 3139396, 12526605, 49902440, 198499200, 788795924, 3131945190, 12428258796, 49295766000, 195464345440, 774857314042, 3071175790232, 12171403236288, 48233597481200, 191138095393700, 757436171945952 (list; graph; refs; listen; history; text; internal format)
 OFFSET 5,2 COMMENTS In other words, the number of (n-5)-dissections of an n-gon modulo the cyclic action. Equivalently, the number of two-dimensional faces of the (n-3)-dimensional associahedron modulo the cyclic action. 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 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 5..200 D. Bowman and A. Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv:1209.6270 [math.CO], 2012. P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601. C. R. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388. FORMULA See Maple program. MAPLE C:=n->binomial(2*n, n)/(n+1); T31:=proc(n) local t1; global C; t1 :=  (n-3)^2*(n-4)*C(n-2)/(4*n*(2*n-5)); if n mod 5 = 0 then t1:=t1+(4/5)*C(n/5-1) fi; if n mod 2 = 0 then t1:=t1+(n-4)*C(n/2-1)/8 fi; t1; end; [seq(T31(n), n=5..40)]; MATHEMATICA 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 *) PROG (PARI) \\ See A295495 for DissectionsModCyclic() { 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 CROSSREFS A diagonal of A295633. Cf. A003444, A003450, A220881. Cf. A003443, A003448, A003450. Sequence in context: A087971 A127919 A074092 * A181326 A220964 A231125 Adjacent sequences:  A003442 A003443 A003444 * A003446 A003447 A003448 KEYWORD nonn AUTHOR EXTENSIONS Entry revised (following Bowman and Regev) by N. J. A. Sloane, Dec 28 2012 Name clarified by Andrew Howroyd, Nov 25 2017 STATUS approved

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Last modified July 9 09:40 EDT 2020. Contains 335542 sequences. (Running on oeis4.)