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A178752 a(n) gives the number of conjugacy classes in the permutation group generated by transposition (1 2) and double n-cycle (1 3 5 7 ... 2n-1)(2 4 6 8 ... 2n). This group is a semidirect product formed by a cyclic group acting on an elementary abelian 2-group of rank n by cyclically permuting the factors.

%I A178752 #12 Jan 24 2019 19:10:12
%S A178752 2,5,8,13,16,28,32,56,80,136,208,400,656,1232,2240,4192,7744,14728,
%T A178752 27632,52664,99968,190984,364768,699760,1342256,2582120,4971248,
%U A178752 9588880,18512848,35795104,69273728,134224064,260301632,505301920,981707008
%N A178752 a(n) gives the number of conjugacy classes in the permutation group generated by transposition (1 2) and double n-cycle (1 3 5 7 ... 2n-1)(2 4 6 8 ... 2n). This group is a semidirect product formed by a cyclic group acting on an elementary abelian 2-group of rank n by cyclically permuting the factors.
%H A178752 G. C. Greubel, <a href="/A178752/b178752.txt">Table of n, a(n) for n = 1..1000</a>
%H A178752 J. A. Siehler, <a href="http://www.maa.org/programs/maa-awards/writing-awards/george-polya-awards/the-finite-lamplighter-groups-a-guided-tour">The Finite Lamplighter Groups: A Guided Tour</a>, College Mathematics Journal, Vol. 43, No. 3 (May 2012), pp. 203-211. - From _N. J. A. Sloane_, Oct 05 2012
%F A178752 a(n) = Sum_{k=0..n-1} ( 1/gcd(n,k) 2^s phi(gcd(n,k)/s), s in divisors(gcd(n,k)) ).
%t A178752 a[n_]:= Sum[(1/GCD[n,k])2^s EulerPhi[GCD[n,k]/s], {k, 0, n-1}, {s, Divisors[GCD[n,k]]}];
%K A178752 easy,nonn
%O A178752 1,1
%A A178752 _Jacob A. Siehler_, Jun 09 2010
%E A178752 More terms from _Robert G. Wilson v_, Jun 10 2010