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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).
10
1, 6, 33, 285, 2115, 16533, 126501, 972441, 7451679, 57167260, 438644841, 3369276867, 25905339483, 199408447446, 1536728368389, 11856420991413, 91579955286519, 708146055343668, 5481535740059577, 42473608898628639
OFFSET
3,2
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.
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
Sequence in context: A024079 A228640 A228618 * A343567 A306182 A354888
KEYWORD
nonn
AUTHOR
Valery A. Liskovets, Apr 13 2006
STATUS
approved