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A090372 Number of unrooted planar 3-constellations with n triangles.

%I A090372 #18 Jul 24 2018 10:21:17
%S A090372 1,6,22,174,1479,16808,201834,2631594,35965555,512062566,7528425420,
%T A090372 113708935808,1756853846316,27676951028496,443411345677658,
%U A090372 7209139541742750,118738765611199983,1978360119497335826
%N A090372 Number of unrooted planar 3-constellations with n triangles.
%C A090372 These are planar maps with bicolored faces having n black triangular faces and an arbitrary number of white faces of degrees multiple to 3. The vertices can be and are colored so that any black triangle is colored counterclockwise 1,2,3. Isomorphisms are required to respect the colorings. Also unrooted bi-Eulerian maps with bicolored both vertices and faces and with 2n edges; the maps are considered up to color-preserve isomorphism.
%H A090372 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 A090372 V. A. Liskovets, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i1r88">Enumerative formulas for unrooted planar maps: a pattern</a>, Electron. J. Combin., 11:1 (2004), R88.
%p A090372 with(numtheory): C_3 := proc(n) local s,d; if n=0 then RETURN(1) else s := -3^n*binomial(3*n,n); for d in divisors(n) do s := s+phi(n/d)*3^d*binomial(3*d,d) od; RETURN((4/(3*n))*(3^n*binomial(3*n,n)/((2*n+1)*(2*n+2))+s/2)); fi; end;
%t A090372 a[0] = 1; a[n_] := Module[{s, d}, s = -3^n Binomial[3n, n]; Do[s = s + EulerPhi[n/d] 3^d Binomial[3d, d], {d, Divisors[n]}]; (4/(3n)) (3^n Binomial[3n, n]/((2n+1)(2n+2)) + s/2)];
%t A090372 Array[a, 18] (* _Jean-François Alcover_, Jul 24 2018, from Maple *)
%Y A090372 Cf. A069729, A090373.
%K A090372 easy,nonn
%O A090372 1,2
%A A090372 _Valery A. Liskovets_, Dec 01 2003