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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

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(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]}];

Array[a, 21] (* Jean-Fran├žois 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: A005772 A233083 A053984 * A295105 A295106 A295107

Adjacent sequences:  A113178 A113179 A113180 * A113182 A113183 A113184

KEYWORD

nonn

AUTHOR

Valery A. Liskovets, Oct 19 2005

EXTENSIONS

More terms from Michel Marcus, Oct 14 2015

STATUS

approved

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Last modified August 3 13:39 EDT 2020. Contains 336198 sequences. (Running on oeis4.)