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 A051695 Number of degree-n even permutations of order exactly 4. 17

%I

%S 0,0,0,0,0,90,630,3780,18900,94500,457380,3825360,31505760,312432120,

%T 2704501800,22984481520,179863997040,1531709328240,13078616488560,

%U 147223414987200,1657733805020160,20131890668255520,226464779237447520,2542924546378413120,27053572399079688000

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

%H Andrew Howroyd, <a href="/A051695/b051695.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) = (A001473(n) + A051685(n))/2.

%F E.g.f.: (exp(x + x^2/2 + x^4/4) + exp(x - x^2/2 - x^4/4) - exp(x + x^2/2) - exp(x - x^2/2))/2. - _Andrew Howroyd_, Feb 01 2020

%t m = 26; ((Exp[x + x^2/2 + x^4/4] + Exp[x - x^2/2 - x^4/4] - Exp[x + x^2/2] - Exp[x - x^2/2])/2 + O[x]^m // CoefficientList[#, x]& // Rest) * Range[m - 1]! (* _Jean-François Alcover_, Feb 09 2020, after _Andrew Howroyd_ *)

%o (PARI) seq(n)={my(A=O(x*x^n)); Vec(serlaplace(exp(x + x^2/2 + x^4/4 + A) + exp(x - x^2/2 - x^4/4 + A) - exp(x + x^2/2 + A) - exp(x - x^2/2 + A))/2, -n)} \\ _Andrew Howroyd_, Feb 01 2020

%Y Cf. A001473, A048099, A051685.

%K easy,nonn

%O 1,6