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A181950 Weighted sum of all cyclic subgroups of the Alternating Group A_n. 3

%I

%S 1,1,4,19,91,571,4096,38599,370399,3771751,40020916,486887611,

%T 6457566259,97397627419,1566407932636,25622476773391,416792928270751,

%U 7346982309720079,141863542111338124,2968348473040595971,65223378275792128771,1460499016109864574691,32600807940657384926884

%N Weighted sum of all cyclic subgroups of the Alternating Group A_n.

%C Sum of the order of all cyclic subgroups of Alt_n.

%C Each permutation is counted as many times as it appears in a cyclic subgroup.

%C a(7) = 2^12 is remarkable as a power of 2.

%H Andrew Howroyd, <a href="/A181950/b181950.txt">Table of n, a(n) for n = 1..50</a>

%F a(n) = Sum_{k=1..A051593(n)} k*A303728(n, k). - _Andrew Howroyd_, Jul 03 2018

%e a(5) = 1*1 + 2*15 + 3*10 + 5*6 = 1 + 30 +30 +30 = 91.

%o (PARI) \\ permcount is number of permutations of given type.

%o permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}

%o a(n)={my(s=0); forpart(p=n, if(sum(i=1,#p,p[i]-1)%2==0, my(d=lcm(Vec(p))); s+=d*permcount(p)/eulerphi(d))); s} \\ _Andrew Howroyd_, Jul 03 2018

%Y Cf. A051593, A051636, A303728.

%K nonn

%O 1,3

%A _Olivier GĂ©rard_, Apr 03 2012

%E Terms a(9) and beyond from _Andrew Howroyd_, Jul 03 2018

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Last modified February 15 21:12 EST 2019. Contains 320138 sequences. (Running on oeis4.)