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

0, 1, 6, 40, 268, 1801, 12120, 81628, 550040, 3707635, 24997966, 168573824, 1136933488, 7668785996, 51731557296, 348991600660, 2354505179952, 15885669341751, 107183855819490, 723217053276952, 4880016412621148, 32929530655094281

Offset: 0

Author

Nicolas Bonichon, Jun 28 2017

Keywords

• nonn

Links

• N. Bonichon and B. Lévêque, A bijection for essentially 4-connected toroidal triangulations, arXiv preprint arXiv:1707.08191 [cs.DM], 2017.
• N. Bonichon and B. Lévêque, A bijection for essentially 4-connected toroidal triangulations, The Electronic Journal of Combinatorics 26 (2019) P1.13.

Crossrefs

Cf. A001764.

Programs

Maple

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);

Mathematica

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}];
CoefficientList[T[x], x] (* Jean-François Alcover, Nov 16 2018 *)

Formula

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.

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.

From Vaclav Kotesovec, Jun 25 2019: (Start)

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

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)