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

Author

Valery A. Liskovets, Dec 01 2003

Keywords

• easy
• nonn

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

Examples

			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.
		

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.

Crossrefs

Cf. A000257, A069727, A090372, A118094.

Programs

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 *)

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

Extensions

More terms from Michel Marcus, Dec 11 2014