A090372 -id:A090372 - cp's OEIS frontend

A090371 Number of unrooted planar 2-constellations with n digons. Also number of n-edge unrooted planar Eulerian maps with bicolored faces.

1, 3, 6, 20, 60, 291, 1310, 6975, 37746, 215602, 1262874, 7611156, 46814132, 293447817, 1868710728, 12068905911, 78913940784, 521709872895, 3483289035186, 23464708686960, 159346213738020, 1090073011199451, 7507285094455566, 52021636161126702

Offset: 1

Valery A. Liskovets, Dec 01 2003

a(n) is also the number of unrooted planar hypermaps with n darts up to orientation-preserving homeomorphism (darts are semi-edges in the particular case of ordinary maps). - Valery A. Liskovets, Apr 13 2006

			The 3 Eulerian maps with 2 edges are the digon and two figure eight graphs ("8") in which both loops are colored, resp., black or white.

R. J. Mathar, Table of n, a(n) for n = 1..100
M. Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. v.24 (2000), 337-368.
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]
A. Mednykh and R. Nedela, 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.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
Timothy R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.

Cf. A000257, A069727, A090372, A118094.

A090371 := proc(n)
    local s, d;
    if n=0 then
        1 ;
    else
        s := -2^n*binomial(2*n, n);
        for d in numtheory[divisors](n) do
            s := s+ numtheory[phi](n/d)*2^d*binomial(2*d, d)
        od;
        3/(2*n)*(2^n*binomial(2*n, n)/((n+1)*(n+2))+s/2);
    fi;
end proc:

h0[n_] := 3*2^(n-1)*Binomial[2*n, n]/((n+1)*(n+2)); a[n_] := (h0[n] + DivisorSum[n, If[#>1, EulerPhi[#]*Binomial[n/#+2, 2]*h0[n/#], 0]&])/n; Array[a, 30] (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)

h0(n) = 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2));
a(n) = (h0(n) + sumdiv(n, d, (d>1)*eulerphi(d)*binomial(n/d+2,2)*h0(n/d)))/n; \\ Michel Marcus, Dec 11 2014

More terms from Michel Marcus, Dec 11 2014

A069726 Number of rooted planar bi-Eulerian maps with 2n edges. Bi-Eulerian: all its vertices and faces are of even valency.

Original entry on oeis.org

1, 1, 6, 54, 594, 7371, 99144, 1412802, 21025818, 323686935, 5120138790, 82812679560, 1364498150904, 22839100002036, 387477144862128, 6651170184185802, 115346229450879978, 2018559015390399615, 35610482089433479410, 632770874050702595670, 11317118106279639106530

Offset: 0

Valery A. Liskovets, Apr 07 2002

Also counts rooted planar 3-constellations with n triangles: rooted planar maps with bicolored faces having n black triangular faces and an arbitrary number of white faces of degrees multiple to 3. - Valery A. Liskovets, Dec 01 2003

G. C. Greubel, Table of n, a(n) for n = 0..650
M. Bousquet-Mélou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration, arXiv:math/0504018 [math.CO], 2005.
M. Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. v.24 (2000), 337-368.
V. A. Kazakov, M. Staudacher and Th. Wynter, Character expansion methods for matrix models of dually weighted graphs, arxiv:hep-th/9502132, 1995; Commun. Math. Phys. 177 (1996), 451-468.
V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A000139, A000257, A006402, A090372.

s := 4*(4-81*z)^(1/2): u := 36*I*z^(1/2): a := (s+u)^(1/3): b := (s-u)^(1/3):
gf := 1 + ((b+a)*s + 108*I*z^(1/2)*(b-a) - 32*(9*z+1))/(432*z):
simplify(series(gf, z, 22)): seq(coeff(%, z, n), n = 0..20);
# Peter Luschny, May 19 2024

Join[{1},Table[3^(n-1) Binomial[3n,n+1]/(n(2n+1)),{n,20}]] (* Harvey P. Dale, Oct 18 2013 *)

A069726(n)=if(n,3^(n-1)*binomial(3*n,n+1)/n/(2*n+1),1)  \\ M. F. Hasler, Mar 26 2012

a(n) = 3^(n-1)*A000139(n).
a(0)=1, a(n) = 3^(n-1)*binomial(3n, n+1)/(n(2n+1)) for n >= 1.
G.f.: A(x) = (1 + 3*y - y^2)/3 where 3*x^2*y^3 - y + 1 = 0.
G.f. satisfies A(z) = 1 -47*z +3*z^2 +3*z*(22-9*z)*A(z) +9*z*(9*z-2)*A(z)^2 -81*z^2*A(z)^3.
a(n) ~ 2^(-2*n-1)*3^(4*n-1/2)/(sqrt(Pi)*n^(5/2)). - Ilya Gutkovskiy, Dec 04 2016
D-finite with recurrence 2*(n+1)*(2*n+1)*a(n) -9*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Mar 29 2023
G.f. 1/3 - 2/(27*z) + sqrt(4 - 81*z)*((sqrt(4 - 81*z)/2 + 9*i*sqrt(z)/2)^(1/3) + (sqrt(4 - 81*z)/2 - 9*i*sqrt(z)/2)^(1/3))/(54*z) - (((sqrt(4 - 81*z)/2 + 9*i*sqrt(z)/2)^(1/3) - (sqrt(4 - 81*z)/2 - 9*i*sqrt(z)/2)^(1/3))*i)/(2*sqrt(z)), where i = sqrt(-1). - Karol A. Penson, May 19 2024

Entry revised by Editors of the OEIS, Mar 26 - 27 2012

A090373 Number of unrooted planar 4-constellations with n quadrangles.

Original entry on oeis.org

1, 10, 60, 875, 14600, 303814, 6846180, 165740155, 4221248540, 112001557620, 3071766596524, 86596464513410, 2498536503831640, 73533104142072810, 2201538635362482480, 66907117946947479163, 2060374053699504740000

Offset: 1

Valery A. Liskovets, Dec 01 2003

These are planar maps with bicolored faces having n black quadrangular faces and an arbitrary number of white faces of degrees multiple to 4. The vertices can be and are colored so that any black quadrangle is colored counterclockwise 1,2,3,4. Isomorphisms are required to respect the colorings.

M. Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. v.24 (2000), 337-368.

Cf. A090372, A090374.

with(numtheory): C_4 := proc(n) local s,d; if n=0 then RETURN(1) else s := -4^n*binomial(4*n,n); for d in divisors(n) do s := s+phi(n/d)*4^d*binomial(4*d,d) od; RETURN((5/(4*n))*(4^n*binomial(4*n,n)/((3*n+1)*(3*n+2))+s/2)); fi; end;

a[n_] := Module[{s}, s = -4^n Binomial[4n, n]; Do[s += EulerPhi[n/d] 4^d Binomial[4d, d], {d, Divisors[n]}]; (5/(4n))(4^n Binomial[4n, n]/((3n+1)(3n+2)) + s/2)];
Array[a, 17] (* Jean-François Alcover, Aug 29 2019 *)

a(n) = (5/(4*n))*(4^n*binomial(4*n,n)/((3*n+1)*(3*n+2))+s/2) where s = -4^n* binomial(4*n,n) + Sum_{d|n} (phi(n/d)*4^d*binomial(4*d,d)). - Jean-François Alcover, Aug 29 2019

cp's OEIS Frontend

A090371 Number of unrooted planar 2-constellations with n digons. Also number of n-edge unrooted planar Eulerian maps with bicolored faces.

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A069726 Number of rooted planar bi-Eulerian maps with 2n edges. Bi-Eulerian: all its vertices and faces are of even valency.

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A090373 Number of unrooted planar 4-constellations with n quadrangles.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula