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
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
KEYWORD
easy,nonn
AUTHOR
Valery A. Liskovets, Dec 01 2003
STATUS
approved