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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).

%I A118094 #35 Aug 23 2022 14:01:03
%S A118094 1,6,33,285,2115,16533,126501,972441,7451679,57167260,438644841,
%T A118094 3369276867,25905339483,199408447446,1536728368389,11856420991413,
%U A118094 91579955286519,708146055343668,5481535740059577,42473608898628639
%N 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).
%H A118094 A. Mednykh and R. Nedela, <a href="http://garsia.math.yorku.ca/fpsac06/papers/9_ps_or_pdf.pdf">Counting unrooted hypermaps on closed orientable surface</a>, 18th Intern. Conf. Formal Power Series & Algebr. Comb., Jun 19, 2006, San Diego, California (USA).
%H A118094 A. Mednykh and R. Nedela, <a href="http://dx.doi.org/10.1016/j.disc.2009.03.033">Enumeration of unrooted hypermaps of a given genus</a>, Discr. Math., 310 (2010), 518-526. [From _N. J. A. Sloane_, Dec 19 2009]
%H A118094 Mednykh, A.; Nedela, R. <a href="https://doi.org/10.1007/s10958-017-3555-5">Recent progress in enumeration of hypermaps</a>,  J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 3.
%H A118094 Timothy R. Walsh, <a href="http://www.info2.uqam.ca/~walsh_t/papers/GENERATING NONISOMORPHIC.pdf">Space-efficient generation of nonisomorphic maps and hypermaps</a>
%H A118094 T. R. Walsh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Walsh/walsh3.html">Space-Efficient Generation of Nonisomorphic Maps and Hypermaps</a>, J. Int. Seq. 18 (2015) # 15.4.3
%p A118094 Phi2 := proc(l)
%p A118094     local a,k ;
%p A118094     a := 0 ;
%p A118094     for k in numtheory[divisors](l) do
%p A118094         a := a+numtheory[mobius](l/k)*k^2 ;
%p A118094     end do:
%p A118094     a ;
%p A118094 end proc:
%p A118094 h0 := proc(m)
%p A118094     if type(m,integer) then
%p A118094         binomial(2*m,m)*3*2^(m-1)/(m+1)/(m+2) ;
%p A118094     else
%p A118094         0;
%p A118094     end if;
%p A118094 end proc:
%p A118094 h1 := proc(n)
%p A118094     local a;
%p A118094     a := 0 ;
%p A118094     if n >= 3 and type(n,integer) then
%p A118094         a := add(2^k*(4^(n-2-k)-1)*binomial(n+k,k),k=0..n-3) ;
%p A118094     end if;
%p A118094     a/3 ;
%p A118094 end proc:
%p A118094 A118094 := proc(n)
%p A118094     binomial(n/2+2,4)*h0(n/2) ;
%p A118094     %+2*binomial(n/3+2,3)*h0(n/3) ;
%p A118094     %+6*binomial(n/4+2,3)*h0(n/4) ;
%p A118094     a := %+12*binomial(n/6+2,3)*h0(n/6) ;
%p A118094     for l in numtheory[divisors](n) do
%p A118094         if modp(n,l) = 0 then
%p A118094             a := a+h1(n/l)*Phi2(l) ;
%p A118094         end if;
%p A118094     end do:
%p A118094     a/n ;
%p A118094 end proc:
%p A118094 seq(A118094(n),n=3..14) ; # _R. J. Mathar_, Dec 17 2014
%t A118094 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 *)
%o A118094 (PARI) h0(n) = if(denominator(n)==1, 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2)), 0);
%o A118094 h1(n) = sum(k=0, n-3, (4^(n-2-k)-1)*binomial(n+k, k)<<k)/3;
%o A118094 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
%Y A118094 Cf. A090371, A118093.
%K A118094 nonn
%O A118094 3,2
%A A118094 _Valery A. Liskovets_, Apr 13 2006