cp's OEIS Frontend

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

%I A003446 M1616 #46 Jan 19 2021 11:48:00
%S A003446 0,1,1,2,6,16,52,170,579,1996,7021,24892,89214,321994,1170282,4277352,
%T A003446 15715249,57999700,214939846,799478680,2983699498,11169391168,
%U A003446 41929537871,157807451672,595340479694,2250901216266,8527700012092,32369067177176
%N A003446 Number of nonequivalent dissections of a polygon into n triangles by nonintersecting diagonals rooted at a cell up to rotation and reflection.
%C A003446 Original name: Triangulated (n+2)-gons rooted at one of the triangles.
%C A003446 Also, the total number of atom-rooted polyenoids. - _Sean A. Irvine_, Oct 05 2015
%D A003446 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003446 Vincenzo Librandi, <a href="/A003446/b003446.txt">Table of n, a(n) for n = 0..200</a>
%H A003446 S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="http://dx.doi.org/10.1021/ci00026a012">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
%H A003446 S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="/A002057/a002057.pdf">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
%H A003446 F. Harary, E. M. Palmer, R. C. Read, <a href="/A000108/a000108_20.pdf">On the cell-growth problem for arbitrary polygons, computer printout, circa 1974</a>
%H A003446 F. Harary, E. M. Palmer and R. C. Read, <a href="http://dx.doi.org/10.1016/0012-365X(75)90041-2">On the cell-growth problem for arbitrary polygons</a>, Discr. Math. 11 (1975), 371-389.
%H A003446 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 A003446 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, Table 1.
%H A003446 P. K. Stockmeyer, <a href="http://dx.doi.org/10.1007/BFb0066456">The charm bracelet problem and its applications</a>, 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.
%H A003446 P. J. Stockmeyer, <a href="/A006078/a006078.pdf">The charm bracelet problem and its applications</a>, 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]
%F A003446 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)).
%F A003446 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
%F A003446 a(n) ~ 4^n/(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 13 2013
%t A003446 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. *)
%Y A003446 Column k=3 of A295259.
%K A003446 nonn,easy,nice
%O A003446 0,4
%A A003446 _N. J. A. Sloane_
%E A003446 Name edited by _Andrew Howroyd_, Nov 20 2017