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%I #28 Jun 09 2019 08:30:18
%S 1,3,17,128,1131,11070,116317,1287480,14829188,176250143,2148687567,
%T 26750057584,338939419026,4359422270652,56799490825125,
%U 748414965684808,9959308633462092,133694287642377756,1808762770097970724,24642635223262953600,337856475305856870275
%N a(n) is the number of corner-rooted hexangulations of girth 6 with n inner faces.
%H O. Bernardi and É. Fusy, <a href="http://dx.doi.org/10.1016/j.jcta.2011.08.006">A bijection for triangulations, quadrangulations, pentagulations, etc.</a>, J. Combin. Theory Ser. A 119 (2012), 218-244.
%H O. Bernardi and É. Fusy, <a href="https://arxiv.org/abs/1007.1292">A bijection for triangulations, quadrangulations, pentagulations, etc.</a>, arXiv:1007.1292 [math.CO], 2010-2011.
%H J. Bouttier and E. Guitter, <a href="http://arxiv.org/abs/1305.4816">A note on irreducible maps with several boundaries</a>, arXiv:1305.4816 [math.CO], 2013.
%H William G. Brown, <a href="http://dx.doi.org/10.1112/plms/s3-14.4.746">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964), 746-768.
%H William G. Brown, <a href="http://dx.doi.org/10.4153/CJM-1965-030-1">Enumeration of quadrangular dissections of the disk</a>, Canad. J. Math., 17 (1965) 302-317
%H W. T. Tutte, <a href="http://dx.doi.org/10.4153/CJM-1963-029-x">A Census of Planar Maps</a>, Canad. J. Math. 15 (1963), 249-271.
%F Bouttier-Guittier give an explicit formula.
%F 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
%F a(n) ~ sqrt(152 - 62*sqrt(6)) * (248*sqrt(6)/9 - 52)^n / (3*sqrt(Pi) * n^(5/2)). - _Vaclav Kotesovec_, Jun 09 2019
%e G.f.: x + 3*x^2 + 17*x^3 + 128*x^4 + 1131*x^5 + 11070*x^6 + ...
%t 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 *)
%Y Cf. A179299, A228704.
%K nonn,easy
%O 1,2
%A _Jonathan Vos Post_, Jul 09 2010
%E Edited by _N. J. A. Sloane_, Sep 06 2013
%E More terms from _Franck Maminirina Ramaharo_, Jan 27 2019
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