A000257 Number of rooted bicubic maps: a(n) = (8*n-4)*a(n-1)/(n+2) for n >= 2, a(0) = a(1) = 1. (Formerly M2927 N1175)

1, 1, 3, 12, 56, 288, 1584, 9152, 54912, 339456, 2149888, 13891584, 91287552, 608583680, 4107939840, 28030648320, 193100021760, 1341536993280, 9390758952960, 66182491668480, 469294031831040, 3346270487838720, 23981605162844160, 172667557172477952

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

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 321.

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

Edward A. Bender and E. Rodney Canfield, The number of degree restricted maps on the sphere, SIAM J. Discr. Math., Vol. 7, No. 1 (1994), pp. 9-15.

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.

Jonathan Bloom and Vince Vatter, Two Vignettes On Full Rook Placements, arXiv preprint arXiv:1310.6073 [math.CO], 2013.

Miklós Bóna, 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, and Claire Pennarun, On the number of planar Eulerian orientations, arXiv:1610.09837 [math.CO], 2016.

Nicolas Bonichon, Mireille Bousquet-Mélou, and 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

Valentin Bonzom, Guillaume Chapuy, Maciej Dolega, Enumeration of non-oriented maps via integrability, Alg. Combin. 5 (6) (2022) pp. 1363-1390, A.2.

Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.

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

Mireille Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math., Vol. 24, No. 4 (2000), pp. 337-368.

Frédéric 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, arXiv:hep-th/9602025, 1996; see Eq. C.1.

Wenjie Fang, Bijective link between Chapoton's new intervals and bipartite planar maps, arXiv:2001.04723 [math.CO], 2020.

Alice L.L. Gao, Sergey Kitaev, and Philip B. Zhang, On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.

Juan B. Gil, David Kenepp, and Michael Weiner, Pattern-avoiding permutations by inactive sites, Pennsylvania State University, Altoona (2020).

Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.

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

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

Zhaoxiang Li and Yanpei Liu, Chromatic sums of general maps on the sphere and the projective plane, Discr. Math., Vol. 307, No. 1 (2007), pp. 78-87.

Valery A. Liskovets and Timothy R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., Vol. 282, No. 1-3 (2004), pp. 209-221.

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

Alexander Mednykh and Roman Nedela, Enumeration of unrooted hypermaps of a given genus, Discr. Math., Vol. 310, No. 3 (2010), pp. 518-526. [From N. J. A. Sloane, Dec 19 2009]

Mednykh, A.; Nedela, R. Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016). Table 2

Wojciech Mlotkowski and 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, arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série., arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.

W. T. Tutte, A Census of Planar Maps, Canad. J. Math., Vol. 15 (1963), pp. 249-271.

T. R. S. Walsh, Hypermaps versus bipartite maps, J. Combin. Th., Series B, Vol. 18, No. 2 (1975), pp. 155-163.

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

T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq., Vol. 18 (2015), Article 15.4.3.

Peter G. Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, International Mathematics Research Notices, Volume 2015, Issue 24 (1 January 2015), pp. 13533-13544.

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

D-finite with recurrence (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 2 X 2 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). a(n) = Integral_{x=0..8} x^n*W(x) dx, n=1,2,3,..., where W(x) = sqrt((8-x)^3/x)/(32*Pi). For n=0 the integral is equal to 3/4. This means that a(n) is the n-th moment, n=0,1,2,..., of the 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

A(x) = (1 + 4*y - y^2)/4, where y = C(2*x), C being the g.f. for A000108. - Gheorghe Coserea, Apr 10 2018

From Amiram Eldar, Mar 22 2022: (Start)

Sum_{n>=0} 1/a(n) = 1985/1029 + 1280*arcsin(1/(2*sqrt(2)))/(343*sqrt(7)).

Sum_{n>=0} (-1)^n/a(n) = 341/729 - 1280*arcsinh(1/(2*sqrt(2)))/2187. (End)

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]
(* Second program: *)
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
(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
(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
(Python)
a000257 = [1]
for n in range(1, 25): a000257.append((8*n-4)*a000257[-1]//(n+2))
print(a000257) # Gennady Eremin, Mar 22 2022

Crossrefs

Cf. A069726, A007054, A298358 (rooted).

First row of array A101477.

Sequence in context: A369482 A366097 A074533 * A301418 A224922 A303925

Adjacent sequences: A000254 A000255 A000256 * A000258 A000259 A000260

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved