A317527
Number of edges in the n-alternating group graph.
Original entry on oeis.org
0, 0, 3, 24, 180, 1440, 12600, 120960, 1270080, 14515200, 179625600, 2395008000, 34248614400, 523069747200, 8499883392000, 146459529216000, 2667655710720000, 51218989645824000, 1033983353475072000, 21896118073589760000, 485363950631239680000, 11240007277776076800000
Offset: 1
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[0] cat [Factorial(n)*(n-2)/2: n in [2..25]]; // Vincenzo Librandi, Jul 31 2018
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Join[{0}, Table[n! (n - 2)/2, {n, 2, 20}]]
CoefficientList[Series[x^2/(2 (-1 + x)^2), {x, 0, 19}], x] Range[20]!
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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