A090373 Number of unrooted planar 4-constellations with n quadrangles.

1, 10, 60, 875, 14600, 303814, 6846180, 165740155, 4221248540, 112001557620, 3071766596524, 86596464513410, 2498536503831640, 73533104142072810, 2201538635362482480, 66907117946947479163, 2060374053699504740000

Offset: 1

Author

Valery A. Liskovets, Dec 01 2003

Keywords

• easy
• nonn

Comments

These are planar maps with bicolored faces having n black quadrangular faces and an arbitrary number of white faces of degrees multiple to 4. The vertices can be and are colored so that any black quadrangle is colored counterclockwise 1,2,3,4. Isomorphisms are required to respect the colorings.

Links

• M. Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. v.24 (2000), 337-368.

Crossrefs

Cf. A090372, A090374.

Programs

Maple

with(numtheory): C_4 := proc(n) local s,d; if n=0 then RETURN(1) else s := -4^n*binomial(4*n,n); for d in divisors(n) do s := s+phi(n/d)*4^d*binomial(4*d,d) od; RETURN((5/(4*n))*(4^n*binomial(4*n,n)/((3*n+1)*(3*n+2))+s/2)); fi; end;

Mathematica

a[n_] := Module[{s}, s = -4^n Binomial[4n, n]; Do[s += EulerPhi[n/d] 4^d Binomial[4d, d], {d, Divisors[n]}]; (5/(4n))(4^n Binomial[4n, n]/((3n+1)(3n+2)) + s/2)];
Array[a, 17] (* Jean-François Alcover, Aug 29 2019 *)

Formula

a(n) = (5/(4*n))*(4^n*binomial(4*n,n)/((3*n+1)*(3*n+2))+s/2) where s = -4^n* binomial(4*n,n) + Sum_{d|n} (phi(n/d)*4^d*binomial(4*d,d)). - Jean-François Alcover, Aug 29 2019