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%I #14 May 09 2023 18:25:35
%S 1,3,5,19,62,301,1413,7304,38294,208052,1149018,6466761,36899604,
%T 213245389,1245624985,7345962126,43688266206,261791220038,
%U 1579363550250,9586582997562,58513327318992,358957495385684,2212294939905234
%N Number of torsion pairs in the cluster category of type A_n up to Auslander-Reiten translation.
%C a(n) is also the number of Ptolemy diagrams on n vertices up to rotation.
%C a(n) is the sum over all polygon dissections up to rotation, where each region of size at least four has weight two.
%H Andrew Howroyd, <a href="/A181519/b181519.txt">Table of n, a(n) for n = 3..500</a>
%H Thorsten Holm, Peter Jorgensen, Martin Rubey, <a href="http://arxiv.org/abs/1010.1184">Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type A_n</a>, arXiv:1010.1184v1 [math.RT], 2010
%F 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]
%e 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.
%o (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
%Y Cf. A181517.
%K nonn
%O 3,2
%A _Martin Rubey_, Oct 26 2010
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