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%I A112944 #11 Jul 24 2018 09:52:45
%S A112944 1,2,7,39,308,3013,33300,394340,4878109,62232321,812825244,
%T A112944 10818489817,146250545528,2003199281223,27747288947266,
%U A112944 388087900316025,5474206895126243,77795972452841542,1112947041203866164,16016508647052018408,231727628211887783830,3368855109532696440867
%N 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.
%H A112944 M. Bousquet, G. Labelle and P. Leroux, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00406-9">Enumeration of planar two-face maps</a>, Discrete Math., vol. 222 (2000), 1-25.
%H A112944 Z. C. Gao, V. A. Liskovets and N. C. Wormald, <a href="http://mathstat.carleton.ca/~zgao/PAPER/poles.pdf">Enumeration of unrooted odd-valent regular planar maps</a>, Preprint, 2005.
%F A112944 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.
%e A112944 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.
%t A112944 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]}];
%t A112944 Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Jul 24 2018 *)
%o A112944 (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
%Y A112944 Cf. A005470, A112945, A113181, A113182.
%K A112944 nonn
%O A112944 0,2
%A A112944 _Valery A. Liskovets_, Oct 10 2005
%E A112944 More terms from _Michel Marcus_, Oct 14 2015

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