A090373 - OEIS

%I #20 Aug 29 2019 10:00:33

%S 1,10,60,875,14600,303814,6846180,165740155,4221248540,112001557620,

%T 3071766596524,86596464513410,2498536503831640,73533104142072810,

%U 2201538635362482480,66907117946947479163,2060374053699504740000

%N Number of unrooted planar 4-constellations with n quadrangles.

%C 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.

%H 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.

%F 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

%p 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;

%t 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)];

%t Array[a, 17] (* _Jean-François Alcover_, Aug 29 2019 *)

%Y Cf. A090372, A090374.

%K easy,nonn

%O 1,2

%A _Valery A. Liskovets_, Dec 01 2003