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A000309 Number of rooted planar bridgeless cubic maps with 2n nodes.
(Formerly M3601 N1460)
1, 1, 4, 24, 176, 1456, 13056, 124032, 1230592, 12629760, 133186560, 1436098560, 15774990336, 176028860416, 1990947110912, 22783499599872, 263411369705472, 3073132646563840, 36143187370967040, 428157758086840320 (list; graph; refs; listen; history; text; internal format)



Also counts rooted planar non-separable triangulations with 3n edges. - Valery A. Liskovets, Dec 01 2003

Equivalently, rooted planar loopless triangulations with 2n triangles. - Noam Zeilberger, Oct 06 2016

Description trees of type (2,2) with n edges. (A description tree of type (a,b) is a rooted plane tree where every internal node is labeled by an integer between a and [b + sum of labels of its children], every leaf is labeled a, and the root is labeled [b + sum of labels of its children]. See Definition 1 and Section 5.2 of Cori and Schaeffer 2003.) - Noam Zeilberger, Oct 08 2017

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


C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.

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).


T. D. Noe, Table of n, a(n) for n = 0..100

Marie Albenque, Dominique Poulalhon, A Generic Method for Bijections between Blossoming Trees and Planar Maps, Electron. J. Combin., 22 (2015), #P2.38.

Olivier Bernardi, Bijective counting of Kreweras walks and loopless triangulations, Journal of Combinatorial Theory, Series A 114:5 (2007), 931-956.

Junliang Cai, Yanpei Liu, The enumeration of rooted nonseparable nearly cubic maps, Discrete Math. 207 (1999), no. 1-3, 9--24. MR1710479 (2000g:05074). See (31).

Robert Cori and Gilles Schaeffer, Description trees and Tutte formulas, Theoretical Computer Science 292:1 (2003), 165-183.

S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.

C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)

Hsien-Kuei Hwang, Mihyun Kang, Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.

R. C. Mullin, On counting rooted triangular maps, Canad. J. Math., v.17 (1965), 373-382.

W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.

W. T. Tutte, On the enumeration of four-colored maps, SIAM J. Appl. Math., 17 (1969), 454-460.

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, 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), Rutgers Experimental Math Seminar, Sep 13 2018. Part 2 is vimeo.com/289910554.


a(n) = 2^(n-1) * A000139(n) for n > 0.

a(n) = 4*a(n-1)*binomial(3n, 3) / binomial(2n+2, 3); a(n) = 2^n*(3*n)!/ ( (n+1)!*(2*n+1)! ).

G.f.: (1/(6*x)) * (hypergeom([ -2/3, -1/3],[1/2],(27/2)*x)-1). - Mark van Hoeij, Nov 02 2009

a(n) ~ 3^(3*n+1/2)/(sqrt(Pi)*2^(n+2)*n^(5/2)). - Ilya Gutkovskiy, Oct 06 2016




f[n_] := 2^n(3n)!/((n + 1)!(2n + 1)!); Table[f[n], {n, 0, 19}] (* Robert G. Wilson v, Sep 21 2004 *)

Join[{1}, RecurrenceTable[{a[1]==1, a[n]==4a[n-1] Binomial[3n, 3]/ Binomial[2n+2, 3]}, a[n], {n, 20}]] (* Harvey P. Dale, May 11 2011 *)


(PARI) a(n) = 2^(n+1)*(3*n)!/(n!*(2*n+2)!); \\ Michel Marcus, Aug 09 2014

(MAGMA) [2^(n+1)*Factorial(3*n)/(Factorial(n)*Factorial(2*n+2)): n in [0..20]]; // Vincenzo Librandi, Aug 10 2014


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

Cf. A000139, A000264, A000356, A002005, A006335.

Sequence in context: A215709 A103334 A156017 * A112914 A007846 A139702

Adjacent sequences:  A000306 A000307 A000308 * A000310 A000311 A000312




N. J. A. Sloane and Robert G. Wilson v


Definition clarified by Michael Albert, Oct 24 2008



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Last modified October 22 23:18 EDT 2018. Contains 316518 sequences. (Running on oeis4.)