OFFSET
1,2
LINKS
M. Bousquet, G. Labelle and P. Leroux, Enumeration of planar two-face maps, Discrete Math., vol. 222 (2000), 1-25.
FORMULA
a(n) = binomial(2*n,n)/4 + (1/(4*n))*Sum_{k|2*n} phi(k)*binomial((2*n/k)-1,floor(n/k))^2 where phi(k) is the Euler function A000010.
EXAMPLE
There exist 3 planar maps with two 4-valent vertices: a map with four parallel edges and two different maps with two parallel edges and one loop in each vertex. Therefore a(2)=3.
MATHEMATICA
a[n_] := Binomial[2n, n]/4 + (1/(4n)) Sum[EulerPhi[k] Binomial[2n/k - 1, Floor[n/k]]^2, {k, Divisors[2n]}];
Array[a, 21] (* Jean-François Alcover, Jul 24 2018 *)
PROG
(PARI) a(n) = binomial(2*n, n)/4 + sumdiv(2*n, k, eulerphi(k)* binomial(2*n/k-1, (n\k))^2)/(4*n); \\ Michel Marcus, Oct 14 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Valery A. Liskovets, Oct 19 2005
EXTENSIONS
More terms from Michel Marcus, Oct 14 2015
STATUS
approved