A118094 Numbers of unrooted hypermaps on the torus with n darts up to orientation-preserving homeomorphism (darts are semi-edges in the particular case of ordinary maps).

1, 6, 33, 285, 2115, 16533, 126501, 972441, 7451679, 57167260, 438644841, 3369276867, 25905339483, 199408447446, 1536728368389, 11856420991413, 91579955286519, 708146055343668, 5481535740059577, 42473608898628639

Offset: 3

Author

Valery A. Liskovets, Apr 13 2006

Keywords

• nonn

Links

• 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]
• 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 3.
• 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. 18 (2015) # 15.4.3

Crossrefs

Cf. A090371, A118093.

Programs

Maple

Phi2 := proc(l)
    local a,k ;
    a := 0 ;
    for k in numtheory[divisors](l) do
        a := a+numtheory[mobius](l/k)*k^2 ;
    end do:
    a ;
end proc:
h0 := proc(m)
    if type(m,integer) then
        binomial(2*m,m)*3*2^(m-1)/(m+1)/(m+2) ;
    else
        0;
    end if;
end proc:
h1 := proc(n)
    local a;
    a := 0 ;
    if n >= 3 and type(n,integer) then
        a := add(2^k*(4^(n-2-k)-1)*binomial(n+k,k),k=0..n-3) ;
    end if;
    a/3 ;
end proc:
A118094 := proc(n)
    binomial(n/2+2,4)*h0(n/2) ;
    %+2*binomial(n/3+2,3)*h0(n/3) ;
    %+6*binomial(n/4+2,3)*h0(n/4) ;
    a := %+12*binomial(n/6+2,3)*h0(n/6) ;
    for l in numtheory[divisors](n) do
        if modp(n,l) = 0 then
            a := a+h1(n/l)*Phi2(l) ;
        end if;
    end do:
    a/n ;
end proc:
seq(A118094(n),n=3..14) ; # R. J. Mathar, Dec 17 2014

Mathematica

h0[n_] := If[Denominator[n] == 1, 3*2^(n-1)*Binomial[2*n, n]/((n+1)*(n+2)), 0]; h1[n_] := Sum[(4^(n-2-k)-1)*Binomial[n+k, k]*2^k, {k, 0, n-3}]/3; phi2[n_] := Sum[MoebiusMu[n/d]*d^2, {d, Divisors[n]}]; a[n_] := (Binomial[n/2+2, 4]*h0[n/2] +  2*Binomial[n/3+2, 3]*h0[n/3]+6*Binomial[n/4+2, 3]*h0[n/4] + 12*Binomial[n/6+2, 3]*h0[n/6] + Sum[ phi2[d]*h1[n/d], {d, Divisors[n]}])/n; Table[a[n], {n, 3, 22}] (* Jean-François Alcover, Dec 18 2014, translated from PARI *)

PARI

h0(n) = if(denominator(n)==1, 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2)), 0);
h1(n) = sum(k=0, n-3, (4^(n-2-k)-1)*binomial(n+k, k)<Michel Marcus, Dec 11 2014 ; corrected by Charles R Greathouse IV, Dec 17 2014