A013302 E.g.f.: cosh(log(x+1)-arctanh(x)) (even powers only).

1, 0, 3, 90, 4725, 396900, 49116375, 8428369950, 1917454163625, 558800927685000, 203054287097536875, 90020733946574681250, 47828015945815128148125, 30001210002374944020187500

Offset: 0

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

• nonn

Comments

Number of degree-2n permutations without odd cycles and with even number of even cycles. E.g.f.: (2-x^2)/(2*sqrt(1-x^2)). - Vladeta Jovovic, Aug 10 2007

Examples

			cosh(log(x+1)-arctanh(x)) = 1+3/4!*x^4+90/6!*x^6+4725/8!*x^8+...
		

Crossrefs

Cf. A013299.

Programs

Mathematica

nn=30;Insert[Select[Range[0,nn]!CoefficientList[Series[Cosh[Log[(1/(1-x^2))^(1/2)]],{x,0,nn}],x],#>0&],0,2]  (* Geoffrey Critzer, Mar 10 2013 *)
With[{nn=30},Take[CoefficientList[Series[Cosh[Log[x+1]-ArcTanh[x]],{x,0,nn}],x] Range[ 0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Mar 05 2023 *)

Formula

a(n) ~ (2*n)^(2*n)/exp(2*n). - Vaclav Kotesovec, Oct 19 2013

Conjecture: a(n) +2*(-2*n^2+2*n-3)*a(n-1) +3*(2*n-3)*(2*n-5)*a(n-2)=0. - R. J. Mathar, Oct 05 2014

Conjecture: a(n) = (4*n-5)*[(2*n-3)!!]^2 +(2*n-3)^2*a(n-1).