A181955
Weighted sum of all cyclic subgroups of prime order in the Alternating group.
Original entry on oeis.org
0, 0, 3, 18, 90, 390, 2205, 10878, 45318, 256350, 5530305, 55869330, 865551258, 9892489698, 78223384785, 470010394350, 24530527675230, 409760923017198, 10595007772540113, 160826214447439770, 1585844008081570650, 16787211082925012730, 1362379219330719093273
Offset: 1
Cf.
A181951 (number of such subgroups).
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a(n)={sum(p=2, n, if(isprime(p), sum(k=1, n\p, if(p>2||k%2==0, n!/(k!*(n-k*p)!*p^k)))*p/(p-1)))} \\ Andrew Howroyd, Jul 03 2018
A181966
Sum of the sizes of normalizers of all prime order cyclic subgroups of the symmetric group S_n.
Original entry on oeis.org
0, 2, 12, 72, 480, 4320, 35280, 322560, 3265920, 39916800, 479001600, 6706022400, 93405312000, 1482030950400, 24845812992000, 418455797760000, 7469435990016000, 147254595231744000, 2919482409811968000, 63255452212592640000, 1430546380807864320000
Offset: 1
Cf.
A181954 for the number of such subgroups.
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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
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a:=function(n) local total, perm, g, p, k;
total:= 0; g:= SymmetricGroup(n);
for p in Filtered([2..n], IsPrime) do for k in [1..QuoInt(n,p)] do
perm:=PermList(List([0..p*k-1], i->i - (i mod p) + ((i + 1) mod p) + 1));
total:=total + Size(Normalizer(g, perm)) * Factorial(n) / (p^k * (p-1) * Factorial(k) * Factorial(n-k*p));
od; od;
return total;
end; # Andrew Howroyd, Jul 30 2018
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a(n)={n!*sum(p=2, n, if(isprime(p), n\p))} \\ Andrew Howroyd, Jul 30 2018
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