0,1

The number of rooted non-separable planar maps with n edges. - Valery A. Liskovets (liskov(AT)im.bas-net.by), Mar 17 2005

The shifted sequence starting with a(1): Number of quadrangular dissections of a square, counted by the number of vertices. Rooted, non-separable planar maps with no multiple edges, in which each non-root face has degree 4.

Number of left ternary trees having n nodes (n>=1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 23 2006

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

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

W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15:3 (1963), 526-545.

A. Del Lungo, F. Del Ristoro and J.-G. Penaud, Left ternary trees and non-separable rooted planar maps, Theor. Comp. Sci., 233, 2000, 201-215.

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

O. Guibert, Stack words, ..., Discr. Math., 210 (2000), 71-85.

W. F. Lunnon, Counting polyominoes, pp. 347-372 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.41.

W. T. Tutte, A census of planar maps, Canad. J. Math., 15 (1963), 249-271.

J. West, Sorting twice through a stack. Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991). Theoret. Comput. Sci. 117 (1993), no. 1-2, 303-313.

D. Zeilberger, A proof of Julian West's conjecture ..., Discrete Math., 102 (1992), 85-93.

T. D. Noe, Table of n, a(n) for n=0..200

M. Bousqet-Melou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 713

I. Gessel and G. Xin, The generating function of ternary trees and continued fractions

P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links, p. 11.

2*C(3n, 2n+1)/(n(n+1)), or 2*(3*n)!/((2*n+1)!*((n+1)!)).

Using Stirling's formula in A000142 it easy to get the asymptotic expression a(n) ~ (27/4)^n / (sqrt(Pi*n / 3) * (2n + 1) * (n + 1)). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 13 2001

G.f. A(z) = 2 + zB(z), where B(z) = 1 - 8z + 2z(5-6z)B - 2z^2(1+3z)B^2 - z^4B^3.

G.f.: (2/(3*x)) * (hypergeom([ -2/3, -1/3],[1/2],(27/4)*x)-1) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 02 2009]

A000139 := n->2*(3*n)!/((2*n+1)!*((n+1)!)); [seq(f(i), i=0..30)];

Cf. A000142.

Cf. A000309, A006335, A004677.

Sequence in context: A032085 A032163 A038078 this_sequence A114572 A052621 A131057

Adjacent sequences: A000136 A000137 A000138 this_sequence A000140 A000141 A000142

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).