OFFSET
3,2
COMMENTS
a(n) is also the number of Ptolemy diagrams on n vertices up to rotation.
a(n) is the sum over all polygon dissections up to rotation, where each region of size at least four has weight two.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 3..500
Thorsten Holm, Peter Jorgensen, Martin Rubey, Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type A_n, arXiv:1010.1184v1 [math.RT], 2010
FORMULA
G.f.: (2*Sum_{d>=1} phi(d)*log(1/(1-P(y^d)))/d ) - (1/3)*(P(y)^3+2*P(y^3)) - (1/2)*(3*P(y)^2+P(y^2)) - 2*P(y) + y*P(y) - y^2 where y*P(y) - y^2 is the g.f. of A181517. [corrected by Andrew Howroyd, May 09 2023]
EXAMPLE
For n=5 there are 5 Ptolemy diagrams up to rotation: the pentagon with no diagonal, the pentagon with one diagonal, the pentagon with two noncrossing diagonals, the pentagon with three diagonals and the pentagon with all five diagonals.
PROG
(PARI) seq(n)={my(p=serreverse(x - x^2*(1 + x)/(1 - x) + O(x*x^n)), P(d)=subst(p + O(x^(n\d+1)), x, x^d)); Vec(2*sum(d=1, n, eulerphi(d)/d*log(1/(1-P(d)))) - (P(1)^3 + 2*P(3))/3 - (3*P(1)^2+P(2))/2 - (2 - x)*P(1) - x^2)} \\ Andrew Howroyd, May 09 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Martin Rubey, Oct 26 2010
STATUS
approved