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
Examples
G.f.: x + 3*x^2 + 17*x^3 + 128*x^4 + 1131*x^5 + 11070*x^6 + ...
Links
- 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.
Programs
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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 *)
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
Extensions
Edited by N. J. A. Sloane, Sep 06 2013
More terms from Franck Maminirina Ramaharo, Jan 27 2019