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A000257 Number of rooted bicubic maps: a(n) = (8n-4)*a(n-1)/(n+2).
(Formerly M2927 N1175)
16
1, 1, 3, 12, 56, 288, 1584, 9152, 54912, 339456, 2149888, 13891584, 91287552, 608583680, 4107939840, 28030648320, 193100021760, 1341536993280, 9390758952960, 66182491668480, 469294031831040, 3346270487838720, 23981605162844160, 172667557172477952 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of rooted Eulerian planar maps with n edges. - Valery A. Liskovets, Apr 07 2002

Number of indecomposable 1342-avoiding permutations of length n.

Also counts rooted planar 2-constellations with n digons. - Valery A. Liskovets, 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, Apr 13 2006

Number of "new" intervals in Tamari lattices of size n (see Chapoton paper). - Ralf Stephan, May 08 2007

REFERENCES

L. M. Koganov, V. A. Liskovets, T. R. S. Walsh, Total vertex enumeration in rooted planar maps, Ars Combin. 54 (2000), 149-160.

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).

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.

J. Bloom, V. Vatter, Two Vignettes On Full Rook Placements, arXiv preprint arXiv:1310.6073 [math.CO], 2013.

M. Bona, Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps, arXiv:math/9702223 [math.CO], 1997.

Nicolas Bonichon, Mireille Bousquet-Mélou, Paul Dorbec, Claire Pennarun, On the number of planar Eulerian orientations, arXiv:1610.09837 [math.CO], 2016.

Nicolas Bonichon, Mireille Bousquet-Mélou, Éric Fusy, Baxter permutations and plane bipolar orientations Sem. Lothar. Combin. 61A (2009/10), Art. B61Ah, 29 pp. See Section 8. - N. J. A. Sloane, Mar 27 2014

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

M. Bousquet-Mélou 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, arXiv:math/0602368 [math.CO], 2006.

P. Di Francesco, O. Golinelli and E. Guitter, Meanders and the Temperley-Lieb algebra (see Eq. C.1).

C. Kassel and C. Reutenauer, Algebraicity of the zeta function associated to a matrix over a free group algebra, arXiv:1303.3481 [math.CO], 2013-2014.

Ph. Leroux, A simple symmetry generating operads related to rooted planar m-ary trees and polygonal numbers, arXiv:math/0512437 [math.CO], 2005.

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, Counting unrooted hypermaps on closed orientable surface, 18th Intern. Conf. Formal Power Series & Algebr. Comb., Jun 19, 2006, San Diego, California (USA).

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]

Wojciech Mlotkowski, Karol A. Penson, A Fuss-type family of positive definite sequences, arXiv:1507.07312 [math.PR], 2015.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série., FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.

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.

Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps

FORMULA

a(0) = 1 and a(n) = 3*2^(n-1)*C(n)/(n+2) for n>=1, where C = Catalan (A000108).

a(n) = 2^(n-2) * A007054(n), n>1.

O.g.f.: 1/4 + (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))). - Karol A. Penson, 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. - Karol A. Penson, Jun 04 2004

O.g.f.: 1 + x*2F1( (1, 3/2); (4); 8*x). - Olivier Gérard, Feb 15 2011

(n + 2) a(n) = (8*n - 4) a(n - 1). - Simon Plouffe, Feb 09 2012

O.g.f.: ((1-8*x)^(3/2) + 8*x^2 + 12*x - 1)/(32*x^2) = 1 + x + 3*x^2 + 12*x^3 + 56*x^4 + .... The related generating function 1 + 3*x^2 + 12*x^4 + 56*x^6 + ... is the zeta function associated to a certain 2x2 matrix of noncommuting variables. See Kassel and Reutenauer, Example 5.1. - Peter Bala, Mar 15 2013

a(n) ~ 3*2^(3*n-1) / (sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Mar 10 2014

0 = a(n) * (64*a(n+1) - 28*a(n+2)) + a(n+1) * (12*a(n+1) + a(n+2)) if n>0. - Michael Somos, Apr 03 2014

Integral representation as the n-th moment of the positive function W(x) on (0,8). In Maple notation, a(n) = int(x^n*W(x), x=0..8), n=1,2,3,... , where W(x)=sqrt((8-x)^3/x)/(32*Pi). For n=0 the integral is equal 3/4. This means that a(n) is the n-th moment, n=0,1,2,..., of probability distribution which is a sum of W(x) as the continuous part and an atom at x=0 with weight 1/4 (Dirac(x)/4). This representation is unique as W(x) is the solution of the Hausdorff moment problem. - Karol A. Penson and Wojciech Mlotkowski, Jul 15 2015

G.f. y satisfies: 0 = 16*x^2*y^2 - (8*x^2+12*x-1)*y + x^2+11*x-1. - Gheorghe Coserea, Nov 22 2016

EXAMPLE

G.f. = 1 + x + 3*x^2 + 12*x^3 + 56*x^4 + 288*x^5 + 1584*x^6 + 9152*x^7 + ...

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

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}]] (* Harvey P. Dale, Dec 18 2011 *)

PROG

(PARI)

C(n)=binomial(2*n, n)/(n+1);

a(n)=if(n==0, 1, 3*2^(n-1)*C(n)/(n+2) ); \\ Joerg Arndt, May 04 2013

(PARI) x='x+O('x^66); Vec( ((1-8*x)^(3/2) + 8*x^2 + 12*x - 1)/(32*x^2) ) \\ Joerg Arndt, May 04 2013

(MAGMA) [1] cat [3*2^n*Factorial(2*n)/((2*n^2+6*n+4)*Factorial(n)^2): n in [1.. 25]]; // Vincenzo Librandi, Oct 21 2014

(PARI)

x='x; y='y; Fxy = 16*x^2*y^2 - (8*x^2+12*x-1)*y + x^2+11*x-1;

seq(N) = {

  my(y0 = 1 + O('x^N), y1=0);

  for (k = 1, N,

    y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);

    if (y1 == y0, break()); y0 = y1);

  Vec(y0);

};

seq(24) \\ Gheorghe Coserea, Nov 30 2016

CROSSREFS

Cf. A069726, A007054.

First row of array A101477.

Sequence in context: A192132 A179486 A074533 * A224922 A215252 A180256

Adjacent sequences:  A000254 A000255 A000256 * A000258 A000259 A000260

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 29 15:31 EDT 2017. Contains 284273 sequences.