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%I #21 Apr 03 2017 10:36:06
%S 0,1,3,8,25,78,270,926,3305,11868,43232,158586,586530,2181088,8154710,
%T 30620868,115435625,436654794,1656793374,6303490610,24041649128,
%U 91899730068,352002058402,1350767683698,5192237233602,19989786008160
%N G.f.: Sum_{k >= 1} (phi(k)/k)*log(1-f(x^k)), where f(x) = (1 - sqrt(1 - 4*x)) / (2*x) - 1 is the g.f. for the Catalan numbers (A000108) C_1, C_2, C_3, ...
%C Counts cycles of objects where the individual objects are anything enumerated by the Catalan numbers C_1, C_2, ...
%C The number of unrooted two-face n-edge maps in the plane (planar with a distinguished outside face). - _Valery A. Liskovets_, Mar 17 2005
%D V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
%H Andrew Howroyd, <a href="/A060404/b060404.txt">Table of n, a(n) for n = 0..200</a>
%H P. Flajolet and M. Soria, <a href="http://algo.inria.fr/flajolet/Publications/publist.html">The Cycle Construction</a> In SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
%H V. A. Liskovets and T. R. Walsh, <a href="http://dx.doi.org/10.1016/j.aam.2005.03.006">Counting unrooted maps on the plane</a>, Advances in Applied Math., 36, No.4 (2006), 364-387.
%F a(n) = (1/n) * Sum_{d|n} phi(n/d) * A000346(d-1) for n>0. - _Andrew Howroyd_, Apr 02 2017
%t max = 25; f[x_] := (1 - Sqrt[1 - 4*x])/(2*x) - 1; gf = Sum[(EulerPhi[k]/k)*Log[1 - f[x^k]], {k, 1, max}]; CoefficientList[ Series[-gf, {x, 0, max}], x] (* _Jean-François Alcover_, Jan 21 2013 *)
%o (PARI)
%o a(n) = sumdiv(n, d, eulerphi(n/d)*(2^(2*d-1) - binomial(2*d-1, d)))/n; \\ _Andrew Howroyd_, Apr 02 2017
%Y Cf. A103943.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Apr 05 2001
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