

A112944


Number of unrooted regular oddvalent planar maps with 2 vertices; maps are considered up to orientationpreserving homeomorphisms and the vertices are of valency 2n+1.


4



1, 2, 7, 39, 308, 3013, 33300, 394340, 4878109, 62232321, 812825244, 10818489817, 146250545528, 2003199281223, 27747288947266, 388087900316025, 5474206895126243, 77795972452841542, 1112947041203866164, 16016508647052018408, 231727628211887783830, 3368855109532696440867
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OFFSET

0,2


LINKS

Table of n, a(n) for n=0..21.
M. Bousquet, G. Labelle and P. Leroux, Enumeration of planar twoface maps, Discrete Math., vol. 222 (2000), 125.
Z. C. Gao, V. A. Liskovets and N. C. Wormald, Enumeration of unrooted oddvalent regular planar maps, Preprint, 2005.


FORMULA

a(n) = (1/2)binomial(2n, n) + (1/(4n+2))sum_{k(2n+1)}phi(k)* binomial(2*floor(n/k), floor(n/k))^2, where phi(k) is the Euler function A000010.


EXAMPLE

There exist 2 planar maps with two 3valent vertices: a map with three parallel edges and a map with one loop in each vertex and a link. Therefore a(1)=2.


PROG

(PARI) a(n) = binomial(2*n, n)/2 + sumdiv(2*n+1, k, eulerphi(k)* binomial(2*(n\k), (n\k))^2)/(4*n+2); \\ Michel Marcus, Oct 14 2015


CROSSREFS

Cf. A005470, A112945, A113181, A113182.
Sequence in context: A266310 A032118 A125660 * A060073 A187806 A103365
Adjacent sequences: A112941 A112942 A112943 * A112945 A112946 A112947


KEYWORD

nonn


AUTHOR

Valery A. Liskovets, Oct 10 2005


EXTENSIONS

More terms from Michel Marcus, Oct 14 2015


STATUS

approved



