OFFSET
1,2
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
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.
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Valery A. Liskovets, Dec 01 2003
EXTENSIONS
More terms from Michel Marcus, Dec 11 2014
STATUS
approved