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

1, 1, 9, 105, 2625, 76545, 3440745, 176080905, 12034447425, 922995698625, 87505195602825, 9203114782686825, 1141501848477415425, 155540530213013570625, 24232951756530007115625, 4112826185329479728735625, 781060320618828163499210625

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..280
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 28

Crossrefs

Cf. A001472, A261317, A261381.

Bisection of column k=4 of A261430.

Programs

Magma

m:=40; R:=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

Maple

spec := [S,{S=Set(Union(Cycle(Z,card=2),Cycle(Z,card=4)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

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 *)

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

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

Formula

a(n) = (2n)! * [x^(2n)] exp(x^2/2 + x^4/4).

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.

a(n) = 2^n*Gamma(n+1/2)*A047974(n)/Pi^(1/2). - Mark van Hoeij, Oct 30 2011