A179300 -id:A179300 - cp's OEIS frontend

A179299 Number of corner-rooted pentagulations of girth 5 with 2n+1 inner faces.

1, 5, 121, 4690, 228065, 12673173, 768897585, 49645423227, 3357669088200, 235393836387360, 16981887962145418, 1254065444086727685, 94424981678123285373, 7227272422780512414100, 560989900421822288646265, 44076648941211191411236261, 3500015582480750626266664105

Offset: 0

Jonathan Vos Post, Jul 09 2010

nonn

Olivier Bernardi and Éric Fusy, A bijection for triangulations, quadrangulations, pentagulations, etc, Journal of Combinatorial Theory, Series A 119, 1 (2012) 218-244; arXiv:1007.1292 [math.CO], 2010-2011.
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.

Cf. A069271, A001764, A179300.

k = 34;
{w0, w1, w2, w3} = FixedPoint[Function[{w0, w1, w2, w3}, {w1^2 + w2, w1^3 + 2 w1 w2 + w3, w1^4 + 3 w1^2 w2 + 2 w1 w3 + w2^2, x (1 + w0)^4} + O[x]^k] @@ # &, ConstantArray[0, 4]];
f = w3 - (w0 w3 + 2 w1 w2);
CoefficientList[f, x][[2 ;; ;; 2]]
(* Andrey Zabolotskiy, Jan 17 2022 *)

Entry edited, terms a(5) and beyond added by Andrey Zabolotskiy, Jan 17 2022

A228704 Number of 4-irreducible maps made up of two hexagons and n squares.

Original entry on oeis.org

1, 6, 21, 62, 180, 540, 1683, 5418, 17901, 60390, 207207, 720954, 2537964, 9023328, 32354910, 116873028, 424901655, 1553574330, 5709123135, 21075032250, 78114397680, 290595385080, 1084663520730, 4060907367660, 15246207481050, 57387012171372, 216517061206998

Offset: 0

N. J. A. Sloane, Sep 06 2013

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

Cf. A179300.

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

Bouttier-Guittier give an explicit formula.
a(0) = 1, and a(n) = (6*(2 *(n - 1))!/(n!*(n - 1)!))*2F1(-5, 1 - n, 2*(1 - n); -1) for n >= 1, where 2F1(a, b, c; z) is the hypergeometric function. - Franck Maminirina Ramaharo, Jan 27 2019
a(n) ~ 729 * 2^(2*n - 6) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 09 2019
D-finite with recurrence n*(9*n-29)*(3*n-10)*a(n) -2*(9*n-20)*(2*n-7)*(3*n-7)*a(n-1)=0. - R. J. Mathar, Feb 08 2021

More terms from Franck Maminirina Ramaharo, Jan 27 2019

cp's OEIS Frontend

A179299 Number of corner-rooted pentagulations of girth 5 with 2n+1 inner faces.

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