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

1,2

Comments

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

Links

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.

Example

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.

Maple

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:

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000257, A069727, A090372, A118094.

Sequence in context: A148603 A185172 A173110 * A288817 A168594 A123559

Adjacent sequences: A090368 A090369 A090370 * A090372 A090373 A090374

Keyword

easy,nonn

Author

Valery A. Liskovets, Dec 01 2003

Extensions

More terms from Michel Marcus, Dec 11 2014

Status

approved