A308523 Number of essentially simple rooted toroidal triangulations with n vertices.

0, 1, 10, 97, 932, 8916, 85090, 810846, 7719048, 73431340, 698187400, 6635738209, 63047912372, 598885073788, 5687581936284, 54005562798252, 512728901004816, 4867263839614716, 46199494669833400, 438481077306427924, 4161316466910824272

Offset

0,3

Links

Michael De Vlieger, Table of n, a(n) for n = 0..499

Nicolas Bonichon, Éric Fusy, Benjamin Lévêque, A bijection for essentially 3-connected toroidal maps, arXiv:1907.04016 [math.CO], 2019.

Éric Fusy, Benjamin Lévêque, Orientations and bijections for toroidal maps with prescribed face-degrees and essential girth, arXiv:1807.00522 [math.CO], 2018. See Proposition 25 p. 37.

Formula

G.f.: A/(1-3*A)^2 where A=x(1+A)^4 is the g.f. of A002293.

From Vaclav Kotesovec, Jun 25 2019: (Start)

Recurrence: 81*(n-1)*(3*n - 2)*(3*n - 1)*(24*n - 37)*a(n) = 24*(13824*n^4 - 59328*n^3 + 92832*n^2 - 62278*n + 14653)*a(n-1) - 2048*(2*n - 3)*(4*n - 7)*(4*n - 5)*(24*n - 13)*a(n-2).

a(n) ~ 2^(8*n - 3) / 3^(3*n). (End)

Maple

n:=20:
dev_A := series(RootOf(A-x*(1+A)^4, A), x = 0, n+1):
seq(coeff(series(subs(A=dev_A, A/(1-3*A)^2), x, n+1), x, k), k=0..n);

Mathematica

terms = 21;
A[_] = 0; Do[A[x_] = x (1 + A[x])^4 + O[x]^terms, terms];
CoefficientList[A[x]/(1 - 3 A[x])^2, x] (* Jean-François Alcover, Jun 17 2019 *)

Crossrefs

Cf. A002293, A308524, A308526, A289208, A006422.

Sequence in context: A287831 A288430 A209262 * A190985 A044642 A363186

Adjacent sequences: A308520 A308521 A308522 * A308524 A308525 A308526

Keyword

nonn

Author

Nicolas Bonichon, Jun 05 2019

Status

approved