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A181967
Sum of the sizes of the normalizers of all prime order cyclic subgroups of the alternating group A_n.
3
0, 0, 3, 24, 180, 1440, 12600, 120960, 1270080, 14515200, 179625600, 2634508800, 37362124800, 566658892800, 9807557760000, 167382319104000, 3023343138816000, 57621363351552000, 1155628453883904000, 25545471085854720000, 587545834974658560000, 13488008733331292160000
OFFSET
1,3
COMMENTS
The first 11 terms of this sequence are the same as A317527. - Andrew Howroyd, Jul 30 2018
LINKS
FORMULA
a(n) = n! * (A013939(n) - floor((n + 2)/4)) / 2. - Andrew Howroyd, Jul 30 2018
PROG
(GAP) List([1..7], n->Sum(Filtered( ConjugacyClassesSubgroups( AlternatingGroup(n)), x->IsPrime( Size( Representative(x))) ), x->Size(x)*Size( Normalizer( AlternatingGroup(n), Representative(x))) )); # Andrew Howroyd, Jul 30 2018
(GAP)
a:=function(n) local total, perm, g, p, k;
total:= 0; g:= AlternatingGroup(n);
for p in Filtered([2..n], IsPrime) do for k in [1..QuoInt(n, p)] do
if p>2 or IsEvenInt(k) then
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));
fi;
od; od;
return total;
end; # Andrew Howroyd, Jul 30 2018
(PARI) a(n)={n!*sum(p=2, n, if(isprime(p), if(p==2, n\4, n\p)))/2} \\ Andrew Howroyd, Jul 30 2018
CROSSREFS
Cf. A181951 for the number of such subgroups.
Cf. A181966 is the symmetric group case.
Sequence in context: A073985 A197209 A317527 * A144087 A110347 A213100
KEYWORD
nonn
AUTHOR
Olivier Gérard, Apr 04 2012
EXTENSIONS
Some incorrect conjectures removed by Andrew Howroyd, Jul 30 2018
Terms a(9) and beyond from Andrew Howroyd, Jul 30 2018
STATUS
approved