cp's OEIS Frontend

A090371 Number of unrooted planar 2-constellations with n digons. Also number of n-edge unrooted planar Eulerian maps with bicolored faces.

%I A090371 #41 Sep 11 2023 09:09:40
%S A090371 1,3,6,20,60,291,1310,6975,37746,215602,1262874,7611156,46814132,
%T A090371 293447817,1868710728,12068905911,78913940784,521709872895,
%U A090371 3483289035186,23464708686960,159346213738020,1090073011199451,7507285094455566,52021636161126702
%N A090371 Number of unrooted planar 2-constellations with n digons. Also number of n-edge unrooted planar Eulerian maps with bicolored faces.
%C A090371 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
%H A090371 R. J. Mathar, <a href="/A090371/b090371.txt">Table of n, a(n) for n = 1..100</a>
%H A090371 M. Bousquet-Mélou and G. Schaeffer, <a href="http://dx.doi.org/10.1006/aama.1999.0673">Enumeration of planar constellations</a>, Adv. in Appl. Math. v.24 (2000), 337-368.
%H A090371 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 A090371 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 A090371 A. Mednykh and R. Nedela, <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 2.
%H A090371 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 A090371 Timothy 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.
%e A090371 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.
%p A090371 A090371 := proc(n)
%p A090371     local s, d;
%p A090371     if n=0 then
%p A090371         1 ;
%p A090371     else
%p A090371         s := -2^n*binomial(2*n, n);
%p A090371         for d in numtheory[divisors](n) do
%p A090371             s := s+ numtheory[phi](n/d)*2^d*binomial(2*d, d)
%p A090371         od;
%p A090371         3/(2*n)*(2^n*binomial(2*n, n)/((n+1)*(n+2))+s/2);
%p A090371     fi;
%p A090371 end proc:
%t A090371 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 *)
%o A090371 (PARI) h0(n) = 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2));
%o A090371 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
%Y A090371 Cf. A000257, A069727, A090372, A118094.
%K A090371 easy,nonn
%O A090371 1,2
%A A090371 _Valery A. Liskovets_, Dec 01 2003
%E A090371 More terms from _Michel Marcus_, Dec 11 2014