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Number of degree-n odd permutations of order dividing 12.
2

%I #9 Jul 08 2019 12:30:38

%S 0,0,1,3,12,60,360,2016,11088,73872,602640,4411440,81677376,934435008,

%T 8100473472,104370819840,1448725616640,15823660179456,247231858514688,

%U 3703908371910912,66727356304757760,1124506454958351360,19305439846610835456

%N Number of degree-n odd permutations of order dividing 12.

%H Robert Israel, <a href="/A326242/b326242.txt">Table of n, a(n) for n = 0..482</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.: (1/2)*exp(x + (1/2)*x^2 + (1/3)*x^3 + (1/4)*x^4 + (1/6)*x^6+(1/12)*x^(12)) - (1/2)*exp(x - (1/2)*x^2 + (1/3)*x^3 - (1/4)*x^4 - (1/6)*x^6-(1/12)*x^(12)).

%e For n=3 the a(3)=3 solutions are (1, 2), (2, 3), (1, 3) (permutations in cyclic notation).

%p E:= (1/2)*exp(x + (1/2)*x^2 + (1/3)*x^3 + (1/4)*x^4 + (1/6)*x^6+(1/12)*x^(12)) - (1/2)*exp(x - (1/2)*x^2 + (1/3)*x^3 - (1/4)*x^4 - (1/6)*x^6-(1/12)*x^(12)):

%p S:= series(E,x,31):

%p seq(coeff(S,x,i)*i!,i=0..30); # _Robert Israel_, Jul 08 2019

%t With[{nn = 22}, CoefficientList[Series[1/2 Exp[x + x^2/2 + x^3/3 + x^4/4 + x^6/6 +x^12/12]-1/2 Exp[x - x^2/2 + x^3/3 - x^4/4 - x^6/6 - x^12/12], {x, 0, nn}], x]*Range[0, nn]!]

%Y Cf. A053502, A326242, A000704, A061130, A061131, A061132, A048099, A051695, A061133, A061134, A061135, A326241.

%K nonn

%O 0,4

%A _Luis Manuel Rivera Martínez_, Jul 06 2019