A003446 Number of nonequivalent dissections of a polygon into n triangles by nonintersecting diagonals rooted at a cell up to rotation and reflection. (Formerly M1616)

0, 1, 1, 2, 6, 16, 52, 170, 579, 1996, 7021, 24892, 89214, 321994, 1170282, 4277352, 15715249, 57999700, 214939846, 799478680, 2983699498, 11169391168, 41929537871, 157807451672, 595340479694, 2250901216266, 8527700012092, 32369067177176

Offset

0,4

Comments

Original name: Triangulated (n+2)-gons rooted at one of the triangles.

Also, the total number of atom-rooted polyenoids. - Sean A. Irvine, Oct 05 2015

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

F. Harary, E. M. Palmer, R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974

F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.

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

Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388, Table 1.

P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.

P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]

Formula

Let c(x) = (1-sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers (A000108), let d(x) = 1+x*c(x^2). Then g.f. is (x/6)*(c^3+2*subs(x=x^3, c)+3*d*subs(x=x^2, c)).

Recurrence: n*(n+1)*(n+2)*(12*n^10 - 396*n^9 + 5713*n^8 - 47417*n^7 + 250708*n^6 - 883176*n^5 + 2104831*n^4 - 3368071*n^3 + 3489712*n^2 - 2133004*n + 587808)*a(n) = 2*(n-1)*n*(n+1)*(24*n^10 - 756*n^9 + 10262*n^8 - 78647*n^7 + 374743*n^6 - 1154043*n^5 + 2323495*n^4 - 3057578*n^3 + 2632172*n^2 - 1456776*n + 412560)*a(n-1) + 4*(n-1)*n*(12*n^11 - 384*n^10 + 5377*n^9 - 43234*n^8 + 219811*n^7 - 731024*n^6 + 1576767*n^5 - 2055172*n^4 + 1195025*n^3 + 527398*n^2 - 1223056*n + 534240)*a(n-2) - 2*(72*n^13 - 2484*n^12 + 37950*n^11 - 339019*n^10 + 1971954*n^9 - 7887993*n^8 + 22425262*n^7 - 46437513*n^6 + 71577166*n^5 - 83189763*n^4 + 71509420*n^3 - 41716412*n^2 + 13543200*n - 1451520)*a(n-3) - 4*(n-1)*n*(2*n - 7)*(24*n^10 - 756*n^9 + 10262*n^8 - 78647*n^7 + 374743*n^6 - 1154043*n^5 + 2323495*n^4 - 3057578*n^3 + 2632172*n^2 - 1456776*n + 412560)*a(n-4) - 8*(n-1)*(2*n - 9)*(12*n^11 - 384*n^10 + 5377*n^9 - 43234*n^8 + 219811*n^7 - 731024*n^6 + 1576767*n^5 - 2055172*n^4 + 1195025*n^3 + 527398*n^2 - 1223056*n + 534240)*a(n-5) + 16*(n-6)*(2*n - 11)*(2*n - 9)*(12*n^10 - 276*n^9 + 2689*n^8 - 14529*n^7 + 48009*n^6 - 101629*n^5 + 142510*n^4 - 137838*n^3 + 93836*n^2 - 39760*n + 6720)*a(n-6). - Vaclav Kotesovec, Aug 13 2013

a(n) ~ 4^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

Mathematica

c[x_] = (1 - Sqrt[1 - 4*x])/(2*x); d[x_] = 1 + x*c[x^2]; f[x_] = (x/6)*(c[x]^3 + 2*c[x^3] + 3*d[x]*c[x^2]); CoefficientList[ Series[ f[x], {x, 0, 27}], x] (* Jean-François Alcover, Sep 30 2011, after g.f. *)

Crossrefs

Column k=3 of A295259.

Sequence in context: A367389 A214833 A263593 * A300668 A263594 A263595

Adjacent sequences: A003443 A003444 A003445 * A003447 A003448 A003449

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Name edited by Andrew Howroyd, Nov 20 2017

Status

approved