A130915 Number of permutations in the symmetric group S_n in which cycle lengths are odd and greater than 1.

1, 0, 0, 2, 0, 24, 40, 720, 2688, 42560, 245376, 4072320, 31672960, 569935872, 5576263680, 109492807424, 1290701905920, 27616577064960, 380852962029568, 8845627365089280, 139696582370328576, 3506062524305162240, 62387728088875499520, 1684340707284076756992

Offset: 0

Author

Vladeta Jovovic, Aug 23 2007

Keywords

• easy
• nonn

Examples

			a(3)=2 because we have (123) and (132).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450 (terms n = 1..200 from Vincenzo Librandi)

Crossrefs

Cf. A000246, A000166.

Programs

Maple

g:=exp(-x)*sqrt((1+x)/(1-x)): gser:=series(g,x=0,30): seq(factorial(n)*coeff(gser,x,n),n=0..20); # Emeric Deutsch, Aug 25 2007
# second Maple program:
a:= proc(n) option remember;
      `if`(n<3, 1/2, a(n-2)+a(n-3))*(n-1)*(n-2)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 18 2024

Mathematica

nn=20;Drop[Range[0,nn]!CoefficientList[Series[((1+x)/(1-x))^(1/2)Exp[-x],{x,0,nn}],x],1]  (* Geoffrey Critzer, Dec 15 2012 *)
a[n_] := (-1)^n*Sum[If[n==k, 1, (-1)^(n + k)*(n - k)!*Sum[Sum[2^(j - i)*StirlingS1[j, i]*Binomial[n - k - 1, j - 1]/j!, {j, i, n - k}], {i, 1, n - k}]*Binomial[n, k]], {k, 0, n}]; Flatten[Table[a[n], {n, 1, 20}]] (* Detlef Meya, Jan 18 2024 *)

PARI

my(x='x+O('x^33)); Vec(serlaplace(exp(-x)*sqrt((1+x)/(1-x)))) \\ Joerg Arndt, Jan 18 2024

Formula

E.g.f.: exp(-x)*sqrt((1+x)/(1-x)).

a(n) ~ 2*n^n/exp(n+1). - Vaclav Kotesovec, Oct 08 2013

a(n) = (-1)^n*Sum_{k = 0..n} (1 if n = k, otherwise (-1)^(n + k)*(n - k)!*Sum_{i = 1..n - k} Sum_{j = i..n - k} 2^(j - i)*Stirling1(j, i)*binomial(n - k - 1, j - 1)/j!*binomial(n, k)). - Detlef Meya, Jan 18 2024

a(n) = (n-1)*(n-2)*(a(n-2)+a(n-3)) for n>=3. - Alois P. Heinz, Jan 18 2024

Extensions

More terms from Emeric Deutsch, Aug 25 2007

a(0)=1 prepended by Alois P. Heinz, Jan 18 2024