%I A000139 M1660 N0651
%S A000139 2,1,2,6,22,91,408,1938,9614,49335,260130,1402440,7702632,42975796,
%T A000139 243035536,1390594458,8038677054,46892282815,275750636070,1633292229030,
%U A000139 9737153323590,58392041019795,352044769046880,2132866978427640
%N A000139 Number of 2-stack sortable permutations on n letters.
%C A000139 The number of rooted non-separable planar maps with n edges. - Valery 
               A. Liskovets (liskov(AT)im.bas-net.by), Mar 17 2005
%C A000139 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.
%C A000139 Number of left ternary trees having n nodes (n>=1). - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Jul 23 2006
%D A000139 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000139 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000139 W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 
               15:3 (1963), 526-545.
%D A000139 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.
%D A000139 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 
               2004; p. 714.
%D A000139 O. Guibert, Stack words, ..., Discr. Math., 210 (2000), 71-85.
%D A000139 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.
%D A000139 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Problem 6.41.
%D A000139 W. T. Tutte, A census of planar maps, Canad. J. Math., 15 (1963), 249-271.
%D A000139 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 A000139 D. Zeilberger, A proof of Julian West's conjecture ..., Discrete Math., 
               102 (1992), 85-93.
%H A000139 T. D. Noe, <a href="b000139.txt">Table of n, a(n) for n=0..200</a>
%H A000139 M. Bousqet-Melou and A. Jehanne, <a href="http://arXiv.org/abs/math.CO/
               0504018">Polynomial equations with one catalytic variable, algebraic 
               series and map enumeration</a>
%H A000139 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               713
%H A000139 I. Gessel and G. Xin, <a href="http://arXiv.org/abs/math.CO/0505217">
               The generating function of ternary trees and continued fractions</
               a>
%H A000139 P. Zinn-Justin and J.-B. Zuber, <a href="http://arXiv.org/abs/math-ph/
               0303049">Matrix integrals and the generation and counting of virtual 
               tangles and links</a>, p. 11.
%F A000139 2*C(3n, 2n+1)/(n(n+1)), or 2*(3*n)!/((2*n+1)!*((n+1)!)).
%F A000139 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
%F A000139 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.
%F A000139 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]
%p A000139 A000139 := n->2*(3*n)!/((2*n+1)!*((n+1)!)); [seq(f(i),i=0..30)];
%Y A000139 Cf. A000142.
%Y A000139 Cf. A000309, A006335, A004677.
%Y A000139 Sequence in context: A032085 A032163 A038078 this_sequence A114572 A052621 
               A131057
%Y A000139 Adjacent sequences: A000136 A000137 A000138 this_sequence A000140 A000141 
               A000142
%K A000139 nonn,easy,nice,new
%O A000139 0,1
%A A000139 N. J. A. Sloane (njas(AT)research.att.com).