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A000168
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2*3^n*(2*n)!/(n!*(n+2)!).
(Formerly M1940 N0768)
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5
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1, 2, 9, 54, 378, 2916, 24057, 208494, 1876446, 17399772, 165297834, 1602117468, 15792300756, 157923007560, 1598970451545, 16365932856990, 169114639522230, 1762352559231660, 18504701871932430, 195621134074714260
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of rooted 4-regular planar maps with n vertices.
Also, number of doodles with n crossings, irrespective of the number of loops.
Contribution from Karol A. Penson (penson(AT)lptl.jussieu.fr), 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)
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REFERENCES
| R. Cori and B. Vauquelin, Planar maps are well labeled trees, Canad. J. Math., 33 (1981), 1023-1042.
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.
R. C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, 3 (1967), 103-121.
S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures<a/>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, < a href="http://arxiv.org/ftp/arxiv/papers/0912/0912.0072.pdf"> Une mthode pour obtenir la fonction gnratrice d'une srie. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
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).
W. T. Tutte, A census of planar maps, Canad. J. Math., 15 (1963), 249-271.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 516
M. Bousquet-Melou, Limit laws for embedded trees
M. Bousqet-Melou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration
G. Schaeffer and P. Zinn-Justin, On the asymptotic number of plane curves and alternating knots
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FORMULA
| G.f. satisfies A(z) = 1 - 16z +18zA - 27z^2A^2.
G.f.: F(1/2,1;3;12x). [From Paul Barry (pbarry(AT)wit.ie), Feb 04 2009]
a(n)=2*3^n*A000108(n)/(n+2). [From Paul Barry (pbarry(AT)wit.ie), Feb 04 2009]
(n + 1) a(n) = (12 n - 18) a(n - 1). [From Simon Plouffe (simon.plouffe(AT)gmail.com), Feb 09 2012.
G.f.: 1/54*(-1+18*x+(-(12*x-1)^3)^(1/2))/x^2. [From Simon Plouffe, Feb 09 2012]
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MAPLE
| f:=n->2*3^n*(2*n)!/(n!*(n+2)!);
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MATHEMATICA
| Table[(2*3^n*(2n)!)/(n!(n+2)!), {n, 0, 20}] (* From Harvey P. Dale, Jul 25 2011 *)
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CROSSREFS
| First row of array A102994. Cf. A005470.
Sequence in context: A192131 A073986 A089436 * A127128 A064151 A075679
Adjacent sequences: A000165 A000166 A000167 * A000169 A000170 A000171
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KEYWORD
| nonn,nice,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Integral representation [From Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 02 2010]
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