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A289208 Number of rooted essentially 4-connected toroidal triangulations with n vertices.

%I A289208 #34 Jan 04 2025 13:48:30
%S A289208 0,1,6,40,268,1801,12120,81628,550040,3707635,24997966,168573824,
%T A289208 1136933488,7668785996,51731557296,348991600660,2354505179952,
%U A289208 15885669341751,107183855819490,723217053276952,4880016412621148,32929530655094281
%N A289208 Number of rooted essentially 4-connected toroidal triangulations with n vertices.
%H A289208 N. Bonichon and B. Lévêque, <a href="https://arxiv.org/abs/1707.08191">A bijection for essentially 4-connected toroidal triangulations</a>, arXiv preprint arXiv:1707.08191 [cs.DM], 2017.
%H A289208 N. Bonichon and B. Lévêque, <a href="https://doi.org/10.37236/7897">A bijection for essentially 4-connected toroidal triangulations</a>, The Electronic Journal of Combinatorics 26 (2019) P1.13.
%F A289208 G.f.: x*A/(7*A^2*x - 21*A*x + 9*x + 1) where A = 1+x*A^3 is the g.f. of A001764.
%F A289208 0 = 729*T^3*x^3 + 2700*T^3*x^2 - 848*T^3*x + 756*T^2*x^2 + 64*T^3 - 112*T^2*x + 54*T*x^2 - T*x + x^2 where T is the g.f. of this sequence.
%F A289208 From _Vaclav Kotesovec_, Jun 25 2019: (Start)
%F A289208 a(n) ~ 3^(3*n) / 2^(2*n + 3).
%F A289208 Recurrence: 32*(n-1)*(2*n-1)*(3*n-1)*(7*n-18)*a(n) = 16*(1113*n^4 - 5753*n^3 + 8619*n^2 - 1717*n - 3462)*a(n-1) - 6*(9450*n^4 - 56367*n^3 + 93156*n^2 - 2813*n - 64226)*a(n-2) - 81*(3*n-8)*(3*n-7)*(3*n+2)*(7*n-11)*a(n-3). (End)
%p A289208 n := 30; t := series(RootOf(729*T^3*x^3+2700*T^3*x^2-848*T^3*x +756*T^2*x^2 +64*T^3 -112*T^2*x +54*T*x^2-T*x+x^2, T), x = 0, n+1): seq(coeff(t, x, k), k = 0 .. n);
%t A289208 terms = 22; T[_] = 0; Do[T[x_] = (1/(x (-1 + 54 x)))(-x^2 + 112 x T[x]^2 - 756 x^2 T[x]^2 - 64 T[x]^3 + 848 x T[x]^3 - 2700 x^2 T[x]^3 - 729 x^3 T[x]^3) + O[x]^terms // Normal, {terms}];
%t A289208 CoefficientList[T[x], x] (* _Jean-François Alcover_, Nov 16 2018 *)
%Y A289208 Cf. A001764.
%K A289208 nonn
%O A289208 0,3
%A A289208 _Nicolas Bonichon_, Jun 28 2017