A113181 Number of unrooted two-vertex (or, dually, two-face) regular planar maps of even valency 2n considered up to orientation-preserving homeomorphism.

1, 3, 14, 95, 859, 9130, 106039, 1297295, 16428300, 213388961, 2827645453, 38086408002, 520062618300, 7184570776213, 100256059855188, 1411319038583375, 20021022607979629, 285965560309310708, 4109498933510809561, 59380204746202961953, 862266486434574492404

Offset

1,2

Links

Table of n, a(n) for n=1..21.

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

Cf. A000010, A113182, A112944.

Sequence in context: A233083 A053984 A395336 * A295105 A392204 A295106

Adjacent sequences: A113178 A113179 A113180 * A113182 A113183 A113184

Keyword

nonn,changed

Author

Valery A. Liskovets, Oct 19 2005

Extensions

More terms from Michel Marcus, Oct 14 2015

Status

approved