|
|
A113181
|
|
Number of unrooted two-vertex (or, dually, two-face) regular planar maps of even valency 2n considered up to orientation-preserving homeomorphism.
|
|
3
|
|
|
1, 3, 14, 95, 859, 9130, 106039, 1297295, 16428300, 213388961, 2827645453, 38086408002, 520062618300, 7184570776213, 100256059855188, 1411319038583375, 20021022607979629, 285965560309310708, 4109498933510809561, 59380204746202961953, 862266486434574492404
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
LINKS
|
|
|
FORMULA
|
a(n) = binomial(2n,n)/4 + 1/(4n) Sum_{k|2n} phi(k) binomial((2n/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]}];
|
|
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
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|