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A052501
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Number of permutations sigma such that sigma^5=Id; degree-n permutations of order dividing 5.
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24
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1, 1, 1, 1, 1, 25, 145, 505, 1345, 3025, 78625, 809425, 4809025, 20787625, 72696625, 1961583625, 28478346625, 238536558625, 1425925698625, 6764765838625, 189239120970625, 3500701266525625, 37764092547420625, 288099608198025625
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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REFERENCES
| L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 26
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FORMULA
| The number of degree-n permutations of order exactly p (where p is prime) satisfies a(n) =a(n-1)+(1+a(n-p))*(n-1)!/(n-p)! with a(n)=0 if p>n. Also a(n)=Sum_{j=1 to floor[n/p]}[n!/(j!*(n-p*j)!*(p^j))].
E.g.f.: exp(x+1/5*x^5)
Recurrence: {a(1)=1, a(0)=1, a(2)=1, a(4)=1, a(3)=1, (-n^4-35*n^2-50*n-24-10*n^3)*a(n)+a(n+5)-a(n+4)}
a(n) = a(n-1)+a(n-5)*(n-1)!/(n-5)! = Sum_{j = 0 to floor[n/5]}[n!/(j!*(n-5j)!*(5^j))] = A059593(n)+1
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MAPLE
| spec := [S, {S=Set(Union(Cycle(Z, card=1), Cycle(Z, card=5)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
| max = 23; CoefficientList[ Series[ Exp[x + x^5/5], {x, 0, max}], x]*Range[0, max]! (* From Jean-François Alcover, Feb 15 2012, after e.g.f. *)
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CROSSREFS
| Cf. A000085, A001470, A001472, A053495-A053505, A005388.
Sequence in context: A072471 A017042 A100255 * A193438 A139152 A123014
Adjacent sequences: A052498 A052499 A052500 * A052502 A052503 A052504
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KEYWORD
| nonn,nice,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jan 15 2000; encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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