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A000168 a(n) = 2*3^n*(2*n)!/(n!*(n+2)!).
(Formerly M1940 N0768)
1, 2, 9, 54, 378, 2916, 24057, 208494, 1876446, 17399772, 165297834, 1602117468, 15792300756, 157923007560, 1598970451545, 16365932856990, 169114639522230, 1762352559231660, 18504701871932430, 195621134074714260, 2080697516976506220, 22254416920705240440, 239234981897581334730, 2583737804493878415084 (list; graph; refs; listen; history; text; internal format)



Number of rooted planar maps with n edges. - Don Knuth, Nov 24 2013

Number of rooted 4-regular planar maps with n vertices.

Also, number of doodles with n crossings, irrespective of the number of loops.

From Karol A. Penson, Sep 02 2010: (Start)

Integral representation as n-th moment of a positive function on the (0,12) segment of the x axis.

In Maple notation: a(n)=int(x^n*(4/9)*sqrt(3)*(1-(1/12)*x)^(3/2)/(Pi*sqrt(x)), x=0..12), n=0,1,...

This representation is unique as it is the solution of the Hausdorff moment problem. (End)

Also, the number of distinct underlying shapes of closed normal linear lambda terms of a given size, where the shape of a lambda term abstracts away from its variable binding. [N. Zeilberger, 2015] - N. J. A. Sloane, Sep 18 2016

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

Number of well-labeled trees (Bona, 2015). - N. J. A. Sloane, Dec 25 2018


Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 319, 353.

E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.

V. A. Liskovets, A census of nonisomorphic planar maps, in Algebraic Methods in Graph Theory, Vol. II, ed. L. Lovasz and V. T. Sos, North-Holland, 1981.

V. A. Liskovets, Enumeration of nonisomorphic planar maps, Selecta Math. Sovietica, 4 (No. 4, 1985), 303-323.

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).


G. C. Greubel, Table of n, a(n) for n = 0..925 [Terms 0 to 100 computed by T. D. Noe; terms 101 to 925 by G. C. Greubel, Jan 15 2017]

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

J.-L. Baril, R. Genestier, A. Giorgetti, and A. Petrossian, Rooted planar maps modulo some patterns, Preprint 2016.

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.

M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.

M. Bousquet-Mélou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration, arXiv:math/0504018 [math.CO], 2005.

Sean R. Carrell and Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

R. Cori and B. Vauquelin, Planar maps are well labeled trees, Canad. J. Math., 33 (1981), 1023-1042.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 516

A. Giorgetti, R. Genestier, and V. Senni, Software Engineering and Enumerative Combinatorics, slides from a talk at MAP 2014.

Hsien-Kuei Hwang, Mihyun Kang, and 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.

C. Kassel, On combinatorial zeta functions, Slides from a talk, Potsdam, 2015.

Sergey Kitaev, Anna de Mier, and Marc Noy, On the number of self-dual rooted maps, European J. Combin. 35 (2014), 377--387. MR3090510. See Eq. (1). - N. J. A. Sloane, May 19 2014

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].

V. A. Liskovets, Enumeration of nonisomorphic planar maps, Journal of Graph Theory, Volume 5, Issue 1, pages 115-117, Spring 1981.

Valery A. Liskovets, A reductive technique for enumerating non-isomorphic planar maps, Discrete Math. 156 (1996), no. 1-3, 197--217. MR1405018 (97f:05087) - From N. J. A. Sloane, Jun 03 2012

R. C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, 3 (1967), 103-121.

R. C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, 3 (1967), 103-121. [Annotated scanned copy]

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.

C. Reutenauer and M. Robado, On an algebraicity theorem of Kontsevich, FPSAC 2012, Nagoya, Japan DMTCS proc. AR, 2012, 241-248. - From N. J. A. Sloane, Dec 23 2012

G. Schaeffer and P. Zinn-Justin, On the asymptotic number of plane curves and alternating knots, arXiv:math-ph/0304034, 2003-2004.

W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

Noam Zeilberger, Counting isomorphism classes of beta-normal linear lambda terms, arXiv:1509.07596 [cs.LO], 2015.

Noam Zeilberger, Towards a mathematical science of programming, Preprint 2015.

Noam Zeilberger, Linear lambda terms as invariants of rooted trivalent maps, arXiv preprint arXiv:1512.06751 [cs.LO], 2015.

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), Part 2, Rutgers Experimental Math Seminar, Sep 13 2018.

Noam Zeilberger and Alain Giorgetti, A correspondence between rooted planar maps and normal planar lambda terms, arXiv:1408.5028 [cs.LO], 2014-2015; Logical Methods in Computer Science, vol. 11 (3:22), 2015, pp. 1-39.

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


G.f. A(z) satisfies A(z) = 1 - 16*z + 18*z*A(z) - 27*z^2*A(z)^2.

G.f.: F(1/2,1;3;12x). - Paul Barry, Feb 04 2009

a(n) = 2*3^n*A000108(n)/(n+2). - Paul Barry, Feb 04 2009

D-finite with recurrence: (n + 1) a(n) = (12 n - 18) a(n - 1). - Simon Plouffe, Feb 09 2012

G.f.: 1/54*(-1+18*x+(-(12*x-1)^3)^(1/2))/x^2. - Simon Plouffe, Feb 09 2012

0 = a(n)*(+144*a(n+1) - 42*a(n+2)) + a(n+1)*(+18*a(n+1) + a(n+2)) if n>=0. - Michael Somos, Jan 31 2014

a(n) ~ 2*(12^n)/((n^2+3*n)*sqrt(Pi*n)). - Peter Luschny, Nov 25 2015

E.g.f.: exp(6*x)*(12*x*BesselI(0,6*x) - (1 + 12*x)*BesselI(1,6*x))/(9*x). - Ilya Gutkovskiy, Feb 01 2017


G.f. = 1 + 2*x + 9*x^2 + 54*x^3 + 378*x^4 + 2916*x^5 + 24057*x^6 + 208494*x^7 + ...




Table[(2*3^n*(2n)!)/(n!(n+2)!), {n, 0, 20}] (* Harvey P. Dale, Jul 25 2011 *)

a[ n_] := If[ n < 0, 0, 2 3^n (2 n)!/(n! (n + 2)!)] (* Michael Somos, Nov 25 2013 *)

a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/2, 1, 3, 12 x], {x, 0, n}] (* Michael Somos, Nov 25 2013 *)


(PARI) {a(n) = if( n<0, 0, 2 * 3^n * (2*n)! / (n! * (n+2)!))}; /* Michael Somos, Nov 25 2013 */

(MAGMA) [(2*Catalan(n)*3^n)/(n+2): n in [1..30]]; // Vincenzo Librandi, Sep 04 2014


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

First row of array A101486.

Cf. A005470.

Rooted maps with n edges of genus g for 0 <= g <= 10: this sequence, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.

Sequence in context: A223943 A241125 A089436 * A307442 A222014 A321974

Adjacent sequences:  A000165 A000166 A000167 * A000169 A000170 A000171




N. J. A. Sloane


More terms from Joerg Arndt, Feb 26 2014



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