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 A052501 Number of permutations sigma such that sigma^5=Id; degree-n permutations of order dividing 5. 26
 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; text; internal format)
 OFFSET 0,6 COMMENTS These are the telephone numbers T^(5)_n of [Artioli et al., p. 7]. - Eric M. Schmidt, Oct 12 2017 REFERENCES R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..300 Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017. Tomislav Došlic, Darko Veljan, Logarithmic behavior of some combinatorial sequences, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019). - N. J. A. Sloane, May 01 2012 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 26 M. B. Kutler, C. R. Vinroot, On q-Analogs of Recursions for the Number of Involutions and Prime Order Elements in Symmetric Groups, JIS 13 (2010) #10.3.6. L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168. 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. MAPLE spec := [S, {S=Set(Union(Cycle(Z, card=1), Cycle(Z, card=5)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20); MATHEMATICA max = 23; CoefficientList[ Series[ Exp[x + x^5/5], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Feb 15 2012, after e.g.f. *) CROSSREFS Cf. A000085, A001470, A001472, A053495-A053505, A005388. Column k=5 of A008307. Sequence in context: A017042 A100255 A305269 * A193438 A139152 A123014 Adjacent sequences:  A052498 A052499 A052500 * A052502 A052503 A052504 KEYWORD nonn,nice,easy AUTHOR N. J. A. Sloane, Jan 15 2000; encyclopedia(AT)pommard.inria.fr, Jan 25 2000 STATUS approved

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Last modified March 18 17:51 EDT 2019. Contains 321292 sequences. (Running on oeis4.)