A002005 Number of rooted planar cubic maps with 2n vertices. (Formerly M3646 N1483)

1, 4, 32, 336, 4096, 54912, 786432, 11824384, 184549376, 2966845440, 48855252992, 820675092480, 14018773254144, 242919827374080, 4261707069259776, 75576645116559360, 1353050213048123392, 24428493151359467520, 444370175232646840320, 8138178004138611179520

Offset

0,2

Comments

Equivalently, number of rooted planar triangulations with 2n faces.

The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018

References

R. C. Mullin, E. Nemeth and P. J. Schellenberg, The enumeration of almost cubic maps, pp. 281-295 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited R. C. Mullin et al., 1970.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..1000

Valentin Bonzom, Guillaume Chapuy, Maciej Dolega, Enumeration of non-oriented maps via integrability, Alg. Combin. 5 (6) (2022) p 1363-1390, A.3

Mireille Bousquet-Mélou, Counting planar maps, coloured or uncoloured, 23rd British Combinatorial Conference, Jul 2011, Exeter, United Kingdom. 392, pp.1-50, 2011, London Math. Soc. Lecture Note Ser., hal-00653963. See p.13.

Evgeniy Krasko and Alexander Omelchenko, Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus, the projective plane and the Klein bottle. Sensed maps on the torus, Discrete Mathematics (2019) Vol. 342, Issue 2, 584-599. Also arXiv:1709.03225 [math.CO].

Maxim Krikun, Explicit enumeration of triangulations with multiple boundaries, arXiv:0706.0681 [math.CO], 2007. [Comment from Gheorghe Coserea, Dec 26 2015: the formula in the paper for almost trivalent maps is 2 * 4^(k-1) * (3k)!!/ ((k+1)!*(k+2)!!); however, the exponent of 4 should be k not (k-1) i.e. 2 * 4^k * (3k)!! / ((k+1)!*(k+2)!!)]

Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.

Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030 [math.LO], March 2018 (A revised version of a 2017 conference paper).

Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Part 2, Rutgers Experimental Math Seminar, Sep 13 2018.

Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.

Formula

a(n) = 2^(2*n+1)*(3*n)!!/((n+2)!*n!!). - Sean A. Irvine, May 19 2013

a(n) ~ sqrt(6/Pi) * n^(-5/2) * (12*sqrt(3))^n. - Gheorghe Coserea, Feb 25 2016

G.f.: (96*x - 1 + 2F1(-2/3, -1/3; 1/2; 432*x^2) - 96*x*2F1(-1/6, 1/6; 3/2; 432*x^2))/(192*x^2). - Benedict W. J. Irwin, Aug 07 2016

From Gheorghe Coserea, Jun 13 2017: (Start)

G.f. y(x) satisfies:

x*(1-432*x^2)*deriv(y,x) = 64*x^2*y^2 + (288*x^2 - 64*x - 1)*y + 72*x + 1.

0 = 64*x^3*y^3 + x*(1-96*x)*y^2 + (30*x-1)*y - 27*x + 1.

(End).

D-finite with recurrence (n+2)*(n+1)*a(n) -48*(3*n-2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Feb 08 2021

From Karol A. Penson and Katarzyna Gorska (katarzyna.gorska@ifj.edu.pl), Nov 02 2022: (Start)

a(n) = Integral_{x=0..12*sqrt(3)} x^n*W(x), where

W(x) = (T1(x) + T2(x)) / T3(x), and

T1(x) = -x^(2/3) * (108 + sqrt(3) * sqrt(432 - x^2));

T2(x) = 3^(1/6)*(36+sqrt(3)*sqrt(432-x^2))^(2/3) * (-432+x^2+36*sqrt(3)* sqrt(432-x^2)) / sqrt(432-x^2);

T3(x) = (128*3^(5/6)*Pi*x^(1/3)*(36+sqrt(3)*sqrt(432-x^2))^(1/3)).

This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with the singularity x^(-1/3), and for x > 0 is monotonically decreasing to zero at x = 12*sqrt(3). (End)

a(n) = 2^(3*n + 1)*binomial(n*3/2, n)/((n + 1)*(n + 2)) = A358367(n) / A000217(n + 1). - Peter Luschny, Nov 14 2022

Maple

seq(2*8^n*binomial(n*3/2, n)/((n + 2)*(n + 1)), n = 0..19); # Peter Luschny, Nov 14 2022

Mathematica

Table[2^(2 n + 1) (3 n)!!/((n + 2)! n!!), {n, 0, 20}] (* Vincenzo Librandi, Dec 28 2015 *)
CoefficientList[Series[(-1 + 96 z + Hypergeometric2F1[-2/3, -1/3, 1/2, 432z^2]- 96 z Hypergeometric2F1[-1/6, 1/6, 3/2, 432z^2])/(192 z^2), {z, 0, 10}], z] (* Benedict W. J. Irwin, Aug 07 2016 *)

Prog

(PARI) factorial2(n) = my(x = (2^(n\2)*(n\2)!)); if (n%2, n!/x, x);
a(n) = 2^(2*n+1)*factorial2(3*n)/((n+2)!*factorial2(n));
vector(20, i, a(i-1))
\\ test: y = Ser(vector(201, n, a(n-1))); x*(1-432*x^2)*y' == 64*x^2*y^2 + (288*x^2 - 64*x - 1)*y + 72*x + 1
\\ Gheorghe Coserea, Jun 13 2017

Crossrefs

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.

Column k=0 of A266240.

Cf. A000217, A358367.

Sequence in context: A362676 A272823 A371655 * A123309 A186391 A231446

Adjacent sequences: A002002 A002003 A002004 * A002006 A002007 A002008

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, May 19 2013

Status

approved