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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.

%I A113181 #17 Sep 07 2024 21:10:21
%S A113181 1,3,14,95,859,9130,106039,1297295,16428300,213388961,2827645453,
%T A113181 38086408002,520062618300,7184570776213,100256059855188,
%U A113181 1411319038583375,20021022607979629,285965560309310708,4109498933510809561,59380204746202961953,862266486434574492404
%N A113181 Number of unrooted two-vertex (or, dually, two-face) regular planar maps of even valency 2n considered up to orientation-preserving homeomorphism.
%H A113181 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.
%F A113181 a(n) = binomial(2*n,n)/4 + (1/(4*n))*Sum_{k|2*n} phi(k)*binomial((2*n/k)-1,floor(n/k))^2 where phi(k) is the Euler function A000010.
%e A113181 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.
%t A113181 a[n_] := Binomial[2n, n]/4 + (1/(4n)) Sum[EulerPhi[k] Binomial[2n/k - 1, Floor[n/k]]^2, {k, Divisors[2n]}];
%t A113181 Array[a, 21] (* _Jean-François Alcover_, Jul 24 2018 *)
%o A113181 (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
%Y A113181 Cf. A000010, A113182, A112944.
%K A113181 nonn
%O A113181 1,2
%A A113181 _Valery A. Liskovets_, Oct 19 2005
%E A113181 More terms from _Michel Marcus_, Oct 14 2015