cp's OEIS Frontend

A002005 Number of rooted planar cubic maps with 2n vertices.

%I A002005 M3646 N1483 #99 Mar 02 2023 08:46:44
%S A002005 1,4,32,336,4096,54912,786432,11824384,184549376,2966845440,
%T A002005 48855252992,820675092480,14018773254144,242919827374080,
%U A002005 4261707069259776,75576645116559360,1353050213048123392,24428493151359467520,444370175232646840320,8138178004138611179520
%N A002005 Number of rooted planar cubic maps with 2n vertices.
%C A002005 Equivalently, number of rooted planar triangulations with 2n faces.
%C A002005 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
%D A002005 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.
%D A002005 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002005 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002005 Gheorghe Coserea, <a href="/A002005/b002005.txt">Table of n, a(n) for n = 0..1000</a>
%H A002005 Valentin Bonzom, Guillaume Chapuy, Maciej Dolega, <a href="https://doi.org/10.5802/alco.268">Enumeration of non-oriented maps via integrability</a>, Alg. Combin. 5 (6) (2022) p 1363-1390, A.3
%H A002005 Mireille Bousquet-Mélou, <a href="https://hal.archives-ouvertes.fr/hal-00653963/">Counting planar maps, coloured or uncoloured</a>, 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.
%H A002005 Evgeniy Krasko and Alexander Omelchenko, <a href="https://doi.org/10.1016/j.disc.2018.07.013">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</a>, Discrete Mathematics (2019) Vol. 342, Issue 2, 584-599. Also <a href="https://arxiv.org/abs/1709.03225">arXiv:1709.03225 [math.CO]</a>.
%H A002005 Maxim Krikun, <a href="http://arxiv.org/abs/0706.0681">Explicit enumeration of triangulations with multiple boundaries</a>, 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)!!)]
%H A002005 Noam Zeilberger, <a href="https://arxiv.org/abs/1804.10540">A theory of linear typings as flows on 3-valent graphs</a>, arXiv:1804.10540 [cs.LO], 2018.
%H A002005 Noam Zeilberger, <a href="https://arxiv.org/abs/1803.10080">A Sequent Calculus for a Semi-Associative Law</a>, arXiv preprint 1803.10030 [math.LO], March 2018 (A revised version of a 2017 conference paper).
%H A002005 Noam Zeilberger, <a href="https://vimeo.com/289907363">A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video)</a>, <a href="https://vimeo.com/289910554">Part 2</a>, Rutgers Experimental Math Seminar, Sep 13 2018.
%H A002005 Jian Zhou, <a href="https://arxiv.org/abs/1810.03883">Fat and Thin Emergent Geometries of Hermitian One-Matrix Models</a>, arXiv:1810.03883 [math-ph], 2018.
%F A002005 a(n) = 2^(2*n+1)*(3*n)!!/((n+2)!*n!!). - _Sean A. Irvine_, May 19 2013
%F A002005 a(n) ~ sqrt(6/Pi) * n^(-5/2) * (12*sqrt(3))^n. - _Gheorghe Coserea_, Feb 25 2016
%F A002005 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
%F A002005 From _Gheorghe Coserea_, Jun 13 2017: (Start)
%F A002005 G.f. y(x) satisfies:
%F A002005 x*(1-432*x^2)*deriv(y,x) = 64*x^2*y^2 + (288*x^2 - 64*x - 1)*y + 72*x + 1.
%F A002005 0 = 64*x^3*y^3 + x*(1-96*x)*y^2 + (30*x-1)*y - 27*x + 1.
%F A002005 (End).
%F A002005 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
%F A002005 From _Karol A. Penson_ and Katarzyna Gorska (katarzyna.gorska@ifj.edu.pl), Nov 02 2022: (Start)
%F A002005 a(n) = Integral_{x=0..12*sqrt(3)} x^n*W(x), where
%F A002005 W(x) = (T1(x) + T2(x)) / T3(x), and
%F A002005 T1(x) = -x^(2/3) * (108 + sqrt(3) * sqrt(432 - x^2));
%F A002005 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);
%F A002005 T3(x) = (128*3^(5/6)*Pi*x^(1/3)*(36+sqrt(3)*sqrt(432-x^2))^(1/3)).
%F A002005 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)
%F A002005 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
%p A002005 seq(2*8^n*binomial(n*3/2, n)/((n + 2)*(n + 1)), n = 0..19); # _Peter Luschny_, Nov 14 2022
%t A002005 Table[2^(2 n + 1) (3 n)!!/((n + 2)! n!!), {n, 0, 20}] (* _Vincenzo Librandi_, Dec 28 2015 *)
%t A002005 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 *)
%o A002005 (PARI) factorial2(n) = my(x = (2^(n\2)*(n\2)!)); if (n%2, n!/x, x);
%o A002005 a(n) = 2^(2*n+1)*factorial2(3*n)/((n+2)!*factorial2(n));
%o A002005 vector(20, i, a(i-1))
%o A002005 \\ 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
%o A002005 \\ _Gheorghe Coserea_, Jun 13 2017
%Y A002005 Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
%Y A002005 Column k=0 of A266240.
%Y A002005 Cf. A000217, A358367.
%K A002005 nonn
%O A002005 0,2
%A A002005 _N. J. A. Sloane_
%E A002005 More terms from _Sean A. Irvine_, May 19 2013