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A000257
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Number of rooted bicubic maps: a(n)=(8n-4)a(n-1)/(n+2).
(Formerly M2927 N1175)
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12
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1, 1, 3, 12, 56, 288, 1584, 9152, 54912, 339456, 2149888, 13891584, 91287552, 608583680, 4107939840, 28030648320, 193100021760, 1341536993280, 9390758952960, 66182491668480, 469294031831040, 3346270487838720
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of rooted Eulerian planar maps with n edges. - Valery A. Liskovets (liskov(AT)im.bas-net.by), Apr 07 2002
Number of indecomposable 1342-avoiding permutations of length n.
Also counts rooted planar 2-constellations with n digons. - Valery Liskovets (liskov(AT)im.bas-net.by), Dec 01 2003
a(n) is also the number of rooted planar hypermaps with n darts (darts are semi-edges in the particular case of ordinary maps). - Valery A. Liskovets (liskov(AT)im.bas-net.by), Apr 13 2006
Number of intervals in Tamari lattices of size n (see Chapoton paper). - Ralf Stephan, May 08 2007
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REFERENCES
| L. M. Koganov, V. A. Liskovets, T. R. S. Walsh, Total vertex enumeration in rooted planar maps, Ars Combin. 54 (2000), 149-160.
Z. Li and Y. Liu, Chromatic sums of general maps on the sphere and the projective plane, Discr. Math. 307 (2007), 78-87.
V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.
A. Mednykh and R. Nedela, Enumeration of unrooted hypermaps of a given genus, Discr. Math., 310 (2010), 518-526. [From N. J. A. Sloane, Dec 19 2009]
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.
T. R. S. Walsh, Hypermaps versus bipartite maps. J. Combin. Th., B18 (1975), 155-163.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
E. A. Bender and E. R. Canfield, The number of degree restricted maps on the sphere, SIAM J. Discr. Math., 7 (1994), 9-15.
M. Bona, [math/9702223] Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps
M. Bousquet-Melou, Limit laws for embedded trees, arXiv:0501266
M. Bousquet-Melou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. v.24 (2000), 337-368.
F. Chapoton, Sur le nombre d'intervalles dans les treillis de Tamari
P. Di Francesco, O. Golinelli and E. Guitter, Meanders and the Temperley-Lieb algebra (see Eq. C.1).
Ph. Leroux, A simple symmetry generating operads related to rooted planar m-ary trees and polygonal numbers, arXiv:math.CO/0512437.
A. Mednykh and R. Nedela, Counting unrooted hypermaps on closed orientable surface, 18th Intern. Conf. Formal Power Series & Algebr. Comb., Jun 19, 2006, San Diego, California (USA).
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, 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 méthode pour obtenir la fonction génératrice d'une série. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
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FORMULA
| 3*2^(n-1)*C(n)/(n+2), C = Catalan (A000108).
O.g.f.: (1/8) * ( -(1-8*x)^(1/2) + 16*(1-8*x)^(1/2)*x+1-8*x ) / ((1-8*x)^(1/2)*x*(1+(1-8*x)^(1/2))) [From Karol A. Penson (penson(AT)lptl.jussieu.fr), Jun 04 2004]
E.g.f.: (1/8) * exp(4*x)*(8*BesselI(0, 4*x)*x-BesselI(1, 4*x)-8*BesselI(1, 4*x)*x)/x. - [From Karol A. Penson (penson(AT)lptl.jussieu.fr), Jun 04 2004]
O.g.f.: 1 + x*2F1( (1, 3/2); (4); 8*x) [From Olivier Gerard (olivier.gerard(AT)gmail.com, Feb 15 2011]
(n + 2) a(n) = (8*n - 4) a(n - 1). [From Simon Plouffe, Feb 09 2012]
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MAPLE
| A000257 := proc(n)
option remember;
if n <=1 then
1;
else
4*(2*n-1)*procname(n-1)/(n+2) ;
end if ;
end proc: # R. J. Mathar, Dec 18 2011
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MATHEMATICA
| CoefficientList[
Series[ 1 + x HypergeometricPFQ[{1, 3/2}, {4}, 8 x], {x, 0, 10}], x]
Join[{1}, Table[3*2^(n-1) CatalanNumber[n]/(n+2), {n, 30}]] (* From Harvey P. Dale, Dec 18 2011 *)
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CROSSREFS
| Cf. A069726.
Equals 2^(n-2) * A007054(n), n>1.
First row of array A102544.
Sequence in context: A192132 A179486 A074533 * A180256 A027390 A009499
Adjacent sequences: A000254 A000255 A000256 * A000258 A000259 A000260
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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