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