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

3,2

Links

Table of n, a(n) for n=3..22.

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

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 *)

Prog

(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)<<k)/3;
phi2(n) = sumdiv(n, d, moebius(n/d)*d^2); 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) + sumdiv(n, d, phi2(d)*h1(n/d)))/n; \\ Michel Marcus, Dec 11 2014 ; corrected by Charles R Greathouse IV, Dec 17 2014

Crossrefs

Cf. A090371, A118093.

Sequence in context: A024079 A228640 A228618 * A343567 A306182 A354888

Adjacent sequences: A118091 A118092 A118093 * A118095 A118096 A118097

Keyword

nonn

Author

Valery A. Liskovets, Apr 13 2006

Status

approved