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

1, 6, 22, 174, 1479, 16808, 201834, 2631594, 35965555, 512062566, 7528425420, 113708935808, 1756853846316, 27676951028496, 443411345677658, 7209139541742750, 118738765611199983, 1978360119497335826

Offset: 1

Author

Valery A. Liskovets, Dec 01 2003

Keywords

• easy
• nonn

Comments

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.

Links

• M. Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. v.24 (2000), 337-368.
• V. A. Liskovets, Enumerative formulas for unrooted planar maps: a pattern, Electron. J. Combin., 11:1 (2004), R88.

Crossrefs

Cf. A069729, A090373.

Programs

Maple

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;

Mathematica

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)];
Array[a, 18] (* Jean-François Alcover, Jul 24 2018, from Maple *)