

A003442


Number of nonequivalent dissections of an ngon into (n3) polygons by nonintersecting diagonals rooted at a cell up to rotation.
(Formerly M2002)


5



1, 2, 11, 48, 208, 858, 3507, 14144, 56698, 226100, 898942, 3565920, 14124496, 55887930, 220985795, 873396480, 3450940830, 13633173180, 53855628554, 212750148000, 840496068160, 3320817060132, 13122294166126, 51860761615488
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OFFSET

4,2


COMMENTS

Number of dissections of regular ngon into n3 polygons without reflection and rooted at a cell.  Sean A. Irvine, May 05 2015
The conditions imposed mean that the dissection will always be composed of one quadrilateral and n1 triangles.  Andrew Howroyd, Nov 23 2017


REFERENCES

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 = 4..200
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595601.
C. R. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370388.
Andrey Zabolotskiy, Illustration for n = 4,5,6


EXAMPLE

Case n=5: A pentagon can be dissected into 1 quadrilateral and 1 triangle. Either one of these can be chosen as the root cell so a(n)=2.  Andrew Howroyd, Nov 23 2017


PROG

(PARI)
DissectionsModCyclicRooted(v)={my(n=#v);
my(q=vector(n)); q[1]=serreverse(xsum(i=3, #v, x^i*v[i])/x + O(x*x^n));
for(i=2, n, q[i]=q[i1]*q[1]);
my(vars=variables(q[1]));
my(u(m, r)=substvec(q[r]+O(x^(n\m+1)), vars, apply(t>t^m, vars)));
my(p=O(x*x^n) + sum(i=3, #v, my(c=v[i]); if(c, c*sumdiv(i, d, eulerphi(d)*u(d, i/d))/i)));
vector(n, i, polcoeff(p, i))}
{ my(v=DissectionsModCyclicRooted(apply(i>if(i>=3&&i<=4, y^(i3) + O(y^2)), [1..25]))); apply(p>polcoeff(p, 1), v[4..#v]) } \\ Andrew Howroyd, Nov 22 2017


CROSSREFS

Cf. A003443, A003454, A220881, A295622.
Sequence in context: A019005 A112288 A192699 * A054894 A270662 A139475
Adjacent sequences: A003439 A003440 A003441 * A003443 A003444 A003445


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Sean A. Irvine, May 05 2015
Name clarified by Andrew Howroyd, Nov 22 2017


STATUS

approved



