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

%I A090373 #20 Aug 29 2019 10:00:33
%S A090373 1,10,60,875,14600,303814,6846180,165740155,4221248540,112001557620,
%T A090373 3071766596524,86596464513410,2498536503831640,73533104142072810,
%U A090373 2201538635362482480,66907117946947479163,2060374053699504740000
%N A090373 Number of unrooted planar 4-constellations with n quadrangles.
%C A090373 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 A090373 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 A090373 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 A090373 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 A090373 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 A090373 Array[a, 17] (* _Jean-François Alcover_, Aug 29 2019 *)
%Y A090373 Cf. A090372, A090374.
%K A090373 easy,nonn
%O A090373 1,2
%A A090373 _Valery A. Liskovets_, Dec 01 2003