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A289208 Number of rooted essentially 4-connected toroidal triangulations with n vertices. 3

%I

%S 0,1,6,40,268,1801,12120,81628,550040,3707635,24997966,168573824,

%T 1136933488,7668785996,51731557296,348991600660,2354505179952,

%U 15885669341751,107183855819490,723217053276952,4880016412621148,32929530655094281

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

%H 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 N. Bonichon and B. Lévêque, <a href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i1p13">A bijection for essentially 4-connected toroidal triangulations</a>, The Electronic Journal of Combinatorics 26 (2019) P1.13.

%F 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 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 From _Vaclav Kotesovec_, Jun 25 2019: (Start)

%F a(n) ~ 3^(3*n) / 2^(2*n + 3).

%F 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).

%F (End)

%p 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 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 CoefficientList[T[x], x] (* _Jean-François Alcover_, Nov 16 2018 *)

%Y Cf. A001764.

%K nonn

%O 0,3

%A _Nicolas Bonichon_, Jun 28 2017

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Last modified September 15 16:12 EDT 2019. Contains 327078 sequences. (Running on oeis4.)