cp's OEIS Frontend

A003447 Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals rooted at a cell up to rotation and reflection.

%I A003447 M1772 #29 Jan 19 2021 11:48:00
%S A003447 1,2,7,26,108,434,1765,7086,28384,113092,449582,1783092,7062611,
%T A003447 27944394,110494113,436699670,1725474562,6816591452,26927828642,
%U A003447 106375090796,420248084468,1660408588852,6561147261682,25930381015756,102496390643352,405212762977544
%N A003447 Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals rooted at a cell up to rotation and reflection.
%C A003447 Number of dissections of regular n-gon into n-3 polygons with reflection and rooted at a cell. - _Sean A. Irvine_, May 13 2015
%C A003447 The dissection will always be composed of one quadrilateral and n-4 triangles. - _Andrew Howroyd_, Nov 24 2017
%D A003447 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003447 Andrew Howroyd, <a href="/A003447/b003447.txt">Table of n, a(n) for n = 4..200</a>
%H A003447 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 A003447 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.
%o A003447 (PARI)
%o A003447 DissectionsModDihedralRooted(v)={my(n=#v);
%o A003447 my(q=vector(n)); q[1]=serreverse(x-sum(i=3, #v, x^i*v[i])/x + O(x*x^n));
%o A003447 for(i=2, n, q[i]=q[i-1]*q[1]);
%o A003447 my(vars=variables(q[1]));
%o A003447 my(u(m, r)=substvec(q[r]+O(x^(n\m+1)), vars, apply(t->t^m, vars)));
%o A003447 my(R=sum(i=1, (#v-1)\2, v[2*i+1]*u(2, i)), Q=sum(i=2, #v\2, v[2*i]*u(2, i-1)), T=sum(i=3, #v, my(c=v[i]); if(c, c*sumdiv(i, d, eulerphi(d)*u(d, i/d))/i)));
%o A003447 my(p=O(x*x^n) + (R*(x+R)/(1-Q) + Q*(u(2,1)+(x+R)^2/(1-Q)^2)/2 + T)/2);
%o A003447 vector(n, i, polcoeff(p, i))}
%o A003447 my(v=DissectionsModDihedralRooted(apply(i->if(i>=3&&i<=4,y^(i-3)+O(y^2)),[1..25]))); apply(p->polcoeff(p,1), v[4..#v])
%Y A003447 Cf. A003448, A003452, A003456.
%K A003447 nonn
%O A003447 4,2
%A A003447 _N. J. A. Sloane_
%E A003447 More terms from _Sean A. Irvine_, May 13 2015
%E A003447 Name clarified by _Andrew Howroyd_, Nov 24 2017