

A059838


Number of permutations in the symmetric group S_n that have even order.


4



0, 0, 1, 3, 15, 75, 495, 3465, 29295, 263655, 2735775, 30093525, 370945575, 4822292475, 68916822975, 1033752344625, 16813959537375, 285837312135375, 5214921734397375, 99083512953550125, 2004231846526284375
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OFFSET

0,4


COMMENTS

From Bob Beals: Let P[n] = probability that a random permutation in S_n has odd order. Then P[n] = sum_k P[random perm in S_n has odd order  n is in a cycle of length k] * P[n is in a cycle of length k]. Now P[n is in a cycle of length k] = 1/n; P[random perm in S_n has odd order  k is even] = 0; P[random perm in S_n has odd order  k is odd] = P[ random perm in S_{nk} has odd order]. So P[n] = (1/n) * sum_{k odd} P[nk] = (1/n) P[n1] + (1/n) sum_{k odd and >=3} P[nk] = (1/n)*P[n1] + ((n2)/n)*P[n2] and P[1] = 1, P[2] = 1/2. The solution is: P[n] = (1  1/2) (1  1/4) ... (11/(2*[n/2])).


LINKS

T. D. Noe, Table of n, a(n) for n=0..100


FORMULA

E.g.f.: (1sqrt(1x^2))/(1x).
a(2n) = (2n1)! + (2n1)a(2n1), a(2n+1) = (2n+1)a(2n).
a(n) = n!  A000246(n).  Victor S. Miller


EXAMPLE

A permutation in S_4 has even order iff it is a transposition, a product of two disjoint transpositions or a 4 cycle so a(4) = C(4,2)+ C(4,2)/2 + 3! = 15.


MAPLE

s := series((1sqrt(1x^2))/(1x), x, 21): for i from 0 to 20 do printf(`%d, `, i!*coeff(s, x, i)) od:


MATHEMATICA

a[n_] := a[n] = n!  ((n1)!  a[n1]) * (n+Mod[n, 2]1); a[0] = 0; Table[a[n], {n, 0, 20}](* JeanFrançois Alcover, Nov 21 2011, after Pari *)
With[{nn=20}, CoefficientList[Series[(1Sqrt[1x^2])/(1x), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Aug 05 2015 *)


PROG

(PARI) a(n)=if(n<1, 0, n!((n1)!a(n1))*(n+n%21))
(GAP) List([1..9], n>Length(Filtered(SymmetricGroup(n), x>(Order(x) mod 2)=0)));


CROSSREFS

Cf. A001189, A000246.
Sequence in context: A278398 A000266 A294340 * A079164 A240941 A047015
Adjacent sequences: A059835 A059836 A059837 * A059839 A059840 A059841


KEYWORD

nonn,nice


AUTHOR

Avi Peretz (njk(AT)netvision.net.il), Feb 25 2001


EXTENSIONS

Additional comments and more terms from Victor S. Miller, Feb 25 2001
Further terms and e.g.f. from Vladeta Jovovic, Feb 28 2001


STATUS

approved



