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A052503 Number of permutations sigma of [2n] without fixed points such that sigma^4 = Id.

%I A052503 #35 Sep 08 2022 08:44:59
%S A052503 1,1,9,105,2625,76545,3440745,176080905,12034447425,922995698625,
%T A052503 87505195602825,9203114782686825,1141501848477415425,
%U A052503 155540530213013570625,24232951756530007115625,4112826185329479728735625,781060320618828163499210625
%N A052503 Number of permutations sigma of [2n] without fixed points such that sigma^4 = Id.
%H A052503 G. C. Greubel, <a href="/A052503/b052503.txt">Table of n, a(n) for n = 0..280</a>
%H A052503 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=28">Encyclopedia of Combinatorial Structures 28</a>
%F A052503 a(n) = (2n)! * [x^(2n)] exp(x^2/2 + x^4/4).
%F A052503 D-finite with recurrence a(n) +(-2*n+1)*a(n-1) -2*(n-1)*(2*n-1)*(2*n-3)*a(n-2)=0, with a(0)=1, a(1)=1, a(2)=9. - Corrected by _R. J. Mathar_, Feb 20 2020 to skip zeros.
%F A052503 a(n) = 2^n*Gamma(n+1/2)*A047974(n)/Pi^(1/2). - _Mark van Hoeij_, Oct 30 2011
%p A052503 spec := [S,{S=Set(Union(Cycle(Z,card=2),Cycle(Z,card=4)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052503 With[{nmax = 40}, CoefficientList[Series[Exp[x^2*(2 + x^2)/4], {x, 0, nmax}], x]*(Range[0, nmax])!][[1 ;; -1 ;; 2]] (* _G. C. Greubel_, May 14 2019 *)
%o A052503 (PARI) x='x+O('x^40); v=Vec(serlaplace( exp(x^2/2 + x^4/4) )); vector(#v\2, n, v[2*n-1]) \\ _G. C. Greubel_, May 14 2019
%o A052503 (Magma) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x^2/2 + x^4/4) )); [Factorial(2*n-2)*b[2*n-1]: n in [1..Floor((m-2)/2)]]; // _G. C. Greubel_, May 14 2019
%o A052503 (Sage) m = 40; T = taylor(exp(x^2/2 + x^4/4), x, 0, 2*m+2); [factorial(2*n)*T.coefficient(x, 2*n) for n in (0..m)] # _G. C. Greubel_, May 14 2019
%Y A052503 Cf. A001472, A261317, A261381.
%Y A052503 Bisection of column k=4 of A261430.
%K A052503 easy,nonn
%O A052503 0,3
%A A052503 encyclopedia(AT)pommard.inria.fr, Jan 25 2000