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

1, 1, 2, 3, 9, 20, 75, 262, 1117, 4783, 21971, 102249, 489077, 2370142, 11654465, 57916324, 290693391, 1471341341, 7504177738, 38532692207, 199076194985, 1034236705992, 5400337050086, 28329240333758, 149244907249629

Offset

0,3

Comments

Original name: number of symmetric dissections of a polygon.

Also number of 2-connected outerplanar graphs on n unlabeled nodes. - Steven Finch, Dec 09 2004

References

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

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

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

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 = 0..200

S. R. Finch, Planar graph growth constants.

Steven R. Finch, Planar graph growth constants [Cached copy, with permission of the author]

E. Krasko, A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.

P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.

T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.

T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.

R. C. Read, On general dissections of a polygon, Preprint (1974)

Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.

James Tilley, Stan Wagon, and Eric Weisstein, A Catalog of Facially Complete Graphs, arXiv:2409.11249 [math.CO], 2024. See pp. 7,11.

Mathematica

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)

Prog

(PARI) \\ See A295419 for DissectionsModDihedral().
my(v=DissectionsModDihedral(apply(i->1, [1..30]))); v[3..#v] \\ Andrew Howroyd, Nov 22 2017

Crossrefs

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

Sequence in context: A097075 A036673 A111189 * A015951 A244666 A101531

Adjacent sequences: A001001 A001002 A001003 * A001005 A001006 A001007

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

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

Name clarified by Andrew Howroyd, Nov 22 2017

Status

approved