A112944 Number of unrooted regular odd-valent planar maps with 2 vertices; maps are considered up to orientation-preserving homeomorphisms and the vertices are of valency 2n+1.
1, 2, 7, 39, 308, 3013, 33300, 394340, 4878109, 62232321, 812825244, 10818489817, 146250545528, 2003199281223, 27747288947266, 388087900316025, 5474206895126243, 77795972452841542, 1112947041203866164, 16016508647052018408, 231727628211887783830, 3368855109532696440867
Offset: 0
Keywords
Examples
There exist 2 planar maps with two 3-valent vertices: a map with three parallel edges and a map with one loop in each vertex and a link. Therefore a(1)=2.
Links
- M. Bousquet, G. Labelle and P. Leroux, Enumeration of planar two-face maps, Discrete Math., vol. 222 (2000), 1-25.
- Z. C. Gao, V. A. Liskovets and N. C. Wormald, Enumeration of unrooted odd-valent regular planar maps, Preprint, 2005.
Programs
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Mathematica
a[n_] := (1/2) Binomial[2n, n] + (1/(4n+2)) Sum[EulerPhi[k] Binomial[2 Floor[n/k], Floor[n/k]]^2, {k, Divisors[2n+1]}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 24 2018 *) -
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
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.
Extensions
More terms from Michel Marcus, Oct 14 2015