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Number of degree-n even permutations of order dividing 4.
13

%I #14 Sep 08 2022 08:45:03

%S 1,1,1,1,4,16,136,736,4096,20224,99856,475696,3889216,31778176,

%T 313696384,2709911296,23006784256,179965340416,1532217039616,

%U 13081112406784,147235213351936,1657791879049216,20132199908571136,226466449808367616,2542933338768769024

%N Number of degree-n even permutations of order dividing 4.

%H Alois P. Heinz, <a href="/A061129/b061129.txt">Table of n, a(n) for n = 0..570</a>

%H Lev Glebsky, Melany Licón, Luis Manuel Rivera, <a href="https://arxiv.org/abs/1907.00548">On the number of even roots of permutations</a>, arXiv:1907.00548 [math.CO], 2019.

%F E.g.f.: exp(x)*cosh(x^2/2 + x^4/4).

%t With[{n=30}, CoefficientList[Series[Exp[x]*Cosh[x^2/2 + x^4/4], {x, 0, n}], x]*Range[0, n]!] (* _G. C. Greubel_, Jul 02 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x)*cosh(x^2/2 + x^4/4) )) \\ _G. C. Greubel_, Jul 02 2019

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x)*Cosh(x^2/2 + x^4/4) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jul 02 2019

%o (Sage) m = 30; T = taylor(exp(x)*cosh(x^2/2 + x^4/4), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, Jul 02 2019

%Y Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135.

%K easy,nonn

%O 0,5

%A _Vladeta Jovovic_, Apr 14 2001