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A001004 Number of nonequivalent dissections of an (n+2)-gon by nonintersecting diagonals up to rotation and reflection.
(Formerly M0898 N0339)

%I M0898 N0339

%S 1,1,2,3,9,20,75,262,1117,4783,21971,102249,489077,2370142,11654465,

%T 57916324,290693391,1471341341,7504177738,38532692207,199076194985,

%U 1034236705992,5400337050086,28329240333758,149244907249629

%N Number of nonequivalent dissections of an (n+2)-gon by nonintersecting diagonals up to rotation and reflection.

%C Original name: number of symmetric dissections of a polygon.

%C Also number of 2-connected outerplanar graphs on n unlabeled nodes. - _Steven Finch_, Dec 09 2004

%D Cameron, Peter J. Some treelike objects. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - _N. J. A. Sloane_, Apr 18 2014

%D Guanzhang Hu, Group theory method for enumeration of outerplanar graphs, Acta Math. Appl. Sinica 14 (1998) 381-387.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H Andrew Howroyd, <a href="/A001004/b001004.txt">Table of n, a(n) for n = 0..200</a>

%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Planar graph growth constants</a>.

%H E. Krasko, A. Omelchenko, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i1p17">Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees</a>, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.

%H 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 T. Motzkin, <a href="http://dx.doi.org/10.1090/S0002-9904-1945-08486-9">The hypersurface cross ratio</a>, Bull. Amer. Math. Soc., 51 (1945), 976-984.

%H T. S. Motzkin, <a href="http://dx.doi.org/10.1090/S0002-9904-1948-09002-4 ">Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products</a>, Bull. Amer. Math. Soc., 54 (1948), 352-360.

%H R. C. Read, <a href="/A001004/a001004.pdf">On general dissections of a polygon</a>, Preprint (1974)

%H C. R. Read, <a href="http://dx.doi.org/10.1007/BF03031688">On general dissections of a polygon</a>, Aequat. math. 18 (1978) 370-388.

%t f[x_, n_]:=x+Sum[(1/r)*Binomial[s-2, r-1]*Binomial[r+s-1, s]*x^s, {r, 1, n}, {s, 2, n}]; F[x_, n_]:=Series[((3x^2-2*x*f[x, n]+f[x, n]^2)- (2+2*x+7*x^2-4*x*f[x, n]+2*f[x, n]^2)*f[x^2, n]+ 2*f[x^2, n]^2)/(4*(2*f[x^2, n]-1))+Sum[If[Mod[k, d]==0, EulerPhi[d]*f[x^d, n]^(k/d)/k, 0], {k, 3, n}, {d, 1, k}]/2, {x, 0, n}]; F[x, 22] (Finch)

%o (PARI) \\ See A295419 for DissectionsModDihedral().

%o my(v=DissectionsModDihedral(apply(i->1, [1..30])));v[3..#v] \\ _Andrew Howroyd_, Nov 22 2017

%Y Cf. A003454, A003455, A003456, A005036, A295260.

%K nonn,nice,easy

%O 0,3

%A _N. J. A. Sloane_

%E More terms from Esa Peuha (esa.peuha(AT)helsinki.fi), Oct 21 2005

%E Name clarified by _Andrew Howroyd_, Nov 22 2017

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Last modified January 23 17:38 EST 2019. Contains 319400 sequences. (Running on oeis4.)