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A181966 Sum of the sizes of normalizers of all prime order cyclic subgroups of the symmetric group S_n.

%I A181966 #18 Jul 30 2018 19:09:04
%S A181966 0,2,12,72,480,4320,35280,322560,3265920,39916800,479001600,
%T A181966 6706022400,93405312000,1482030950400,24845812992000,418455797760000,
%U A181966 7469435990016000,147254595231744000,2919482409811968000,63255452212592640000,1430546380807864320000
%N A181966 Sum of the sizes of normalizers of all prime order cyclic subgroups of the symmetric group S_n.
%H A181966 Andrew Howroyd, <a href="/A181966/b181966.txt">Table of n, a(n) for n = 1..200</a>
%H A181966 Math.stackexchange.com, <a href="https://math.stackexchange.com/questions/1218392/normalizer-of-the-cyclic-group-in-s-n">Normalizer of the cyclic group in S_n</a>
%F A181966 a(n) = n! * A013939(n). - _Andrew Howroyd_, Jul 30 2018
%o A181966 (GAP) List([1..7], n->Sum(Filtered( ConjugacyClassesSubgroups( SymmetricGroup(n)), x->IsPrime( Size( Representative(x))) ), x->Size(x)*Size( Normalizer( SymmetricGroup(n), Representative(x))) )); # _Andrew Howroyd_, Jul 30 2018
%o A181966 (GAP)
%o A181966 a:=function(n) local total, perm, g, p, k;
%o A181966   total:= 0; g:= SymmetricGroup(n);
%o A181966   for p in Filtered([2..n], IsPrime) do for k in [1..QuoInt(n,p)] do
%o A181966      perm:=PermList(List([0..p*k-1], i->i - (i mod p) + ((i + 1) mod p) + 1));
%o A181966      total:=total + Size(Normalizer(g, perm)) * Factorial(n) / (p^k * (p-1) * Factorial(k) * Factorial(n-k*p));
%o A181966   od; od;
%o A181966   return total;
%o A181966 end; # _Andrew Howroyd_, Jul 30 2018
%o A181966 (PARI) a(n)={n!*sum(p=2, n, if(isprime(p), n\p))} \\ _Andrew Howroyd_, Jul 30 2018
%Y A181966 Cf. A181954 for the number of such subgroups.
%Y A181966 Cf. A013939, A181967.
%K A181966 nonn
%O A181966 1,2
%A A181966 _Olivier Gérard_, Apr 04 2012
%E A181966 Terms a(8) and beyond from _Andrew Howroyd_, Jul 30 2018