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%I #38 Feb 17 2022 00:55:31
%S 1,1,1,1,1,25,145,505,1345,3025,78625,809425,4809025,20787625,
%T 72696625,1961583625,28478346625,238536558625,1425925698625,
%U 6764765838625,189239120970625,3500701266525625,37764092547420625,288099608198025625
%N Number of permutations sigma such that sigma^5=Id; degree-n permutations of order dividing 5.
%C 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..floor(n/p)} (n!/(j!*(n-p*j)!*(p^j))).
%C These are the telephone numbers T^(5)_n of [Artioli et al., p. 7]. - _Eric M. Schmidt_, Oct 12 2017
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
%H Alois P. Heinz, <a href="/A052501/b052501.txt">Table of n, a(n) for n = 0..300</a>
%H Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, Simonetta Pagnutti, <a href="https://arxiv.org/abs/1703.07262">Motzkin Numbers: an Operational Point of View</a>, arXiv:1703.07262 [math.CO], 2017.
%H Tomislav Došlic, Darko Veljan, <a href="http://dx.doi.org/10.1016/j.disc.2007.04.066">Logarithmic behavior of some combinatorial sequences</a>, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019). - _N. J. A. Sloane_, May 01 2012
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=26">Encyclopedia of Combinatorial Structures 26</a>
%H M. B. Kutler, C. R. Vinroot, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Vinroot/vinroot3.html">On q-Analogs of Recursions for the Number of Involutions and Prime Order Elements in Symmetric Groups</a>, JIS 13 (2010) #10.3.6.
%H L. Moser and M. Wyman, <a href="http://dx.doi.org/10.4153/CJM-1955-020-0">On solutions of x^d = 1 in symmetric groups</a>, Canad. J. Math., 7 (1955), 159-168.
%F E.g.f.: exp(x + x^5/5).
%F a(n+5) = a(n+4) + (24 +50*n +35*n^2 +10*n^3 +n^4)*a(n), with a(0)= ... = a(4) = 1.
%F a(n) = a(n-1) + a(n-5)*(n-1)!/(n-5)!.
%F a(n) = Sum_{j = 0..floor(n/5)} n!/(5^j * j! * (n-5*j)!).
%F a(n) = A059593(n) + 1.
%p spec := [S,{S=Set(Union(Cycle(Z,card=1),Cycle(Z,card=5)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t max = 30; CoefficientList[ Series[ Exp[x + x^5/5], {x, 0, max}], x]*Range[0, max]! (* _Jean-François Alcover_, Feb 15 2012, after e.g.f. *)
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x + x^5/5) )) \\ _G. C. Greubel_, May 14 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^5/5) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, May 14 2019
%o (Sage) m = 30; T = taylor(exp(x + x^5/5), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, May 14 2019
%Y Cf. A000085, A001470, A001472, A053495-A053505, A005388.
%Y Column k=5 of A008307.
%K nonn,nice,easy
%O 0,6
%A _N. J. A. Sloane_, Jan 15 2000; encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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