A179300 a(n) is the number of corner-rooted hexangulations of girth 6 with n inner faces.

1, 3, 17, 128, 1131, 11070, 116317, 1287480, 14829188, 176250143, 2148687567, 26750057584, 338939419026, 4359422270652, 56799490825125, 748414965684808, 9959308633462092, 133694287642377756, 1808762770097970724, 24642635223262953600, 337856475305856870275

Offset

1,2

Links

Table of n, a(n) for n=1..21.

O. Bernardi and É. Fusy, A bijection for triangulations, quadrangulations, pentagulations, etc., J. Combin. Theory Ser. A 119 (2012), 218-244.

O. Bernardi and É. Fusy, A bijection for triangulations, quadrangulations, pentagulations, etc., arXiv:1007.1292 [math.CO], 2010-2011.

J. Bouttier and E. Guitter, A note on irreducible maps with several boundaries, arXiv:1305.4816 [math.CO], 2013.

William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964), 746-768.

William G. Brown, Enumeration of quadrangular dissections of the disk, Canad. J. Math., 17 (1965) 302-317

W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

Formula

Bouttier-Guittier give an explicit formula.

a(1) = 1, and a(n) = (6*(2*(-2 + n))!/((-2 + n)!*n!))*2F1(-5*n, 2 - n, 2*(2 - n); -1) for n >= 2, where 2F1(a, b, c; z) is the hypergeometric function. - Franck Maminirina Ramaharo, Jan 27 2019

a(n) ~ sqrt(152 - 62*sqrt(6)) * (248*sqrt(6)/9 - 52)^n / (3*sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Jun 09 2019

Example

G.f.: x + 3*x^2 + 17*x^3 + 128*x^4 + 1131*x^5 + 11070*x^6 + ...

Mathematica

Join[{1}, Table[(6*(2*(-2 + n))!/((-2 + n)!*n!))*Hypergeometric2F1[-5*n, 2 - n, 2*(2 - n), -1], {n, 2, 50}]] (* Franck Maminirina Ramaharo, Jan 27 2019 *)

Crossrefs

Cf. A179299, A228704.

Sequence in context: A246963 A093460 A187613 * A199493 A245059 A321587

Adjacent sequences: A179297 A179298 A179299 * A179301 A179302 A179303

Keyword

nonn,easy

Author

Jonathan Vos Post, Jul 09 2010

Extensions

Edited by N. J. A. Sloane, Sep 06 2013

More terms from Franck Maminirina Ramaharo, Jan 27 2019

Status

approved