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

1,2

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

Table of n, a(n) for n=1..17.

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

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

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 *)

Crossrefs

Cf. A090372, A090374.

Sequence in context: A281863 A219368 A052664 * A218427 A354944 A041184

Adjacent sequences: A090370 A090371 A090372 * A090374 A090375 A090376

Keyword

easy,nonn

Author

Valery A. Liskovets, Dec 01 2003

Status

approved