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%I M4206 N1756 #24 Sep 08 2022 08:44:29
%S 0,0,0,6,30,180,840,5460,30996,209160,1290960,9753480,69618120,
%T 571627056,4443697440,40027718640,346953934320,3369416698080,
%U 31421601510336,328430320909920,3331475969159520,37124416523261760
%N Number of degree-n permutations of order exactly 4.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A001473/b001473.txt">Table of n, a(n) for n = 1..565</a>
%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^2/2 + x^4/4) - exp(x + x^2/2).
%t Rest@With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^4/4] - Exp[x +x^2/2], {x, 0, m}], x]*Range[0, m]!] (* _G. C. Greubel_, May 14 2019 *)
%o (PARI) my(x=xx+O(xx^33)); concat([0,0,0], Vec(serlaplace(-exp(x+1/2*x^2) +exp(x+1/2*x^2+1/4*x^4)))) \\ _Michel Marcus_, Dec 12 2014
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 +x^4/4) -Exp(x+x^2/2) )); [0,0,0] cat [Factorial(n+3)*b[n]: n in [1..m-4]]; // _G. C. Greubel_, May 14 2019
%o (Sage) m = 30; T = taylor(exp(x +x^2/2 +x^4/4) - exp(x+x^2/2), x, 0, m); a=[factorial(n)*T.coefficient(x, n) for n in (0..m)]; a[1:] # _G. C. Greubel_, May 14 2019
%Y Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.
%Y Column k=4 of A057731.
%K nonn
%O 1,4
%A _N. J. A. Sloane_ and _J. H. Conway_
%E More terms from _Vladeta Jovovic_, Apr 14 2001
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