A092691 a(n) = n! * Sum_{k=1..floor(n/2)} 1/(2k).

0, 0, 1, 3, 18, 90, 660, 4620, 42000, 378000, 4142880, 45571680, 586776960, 7628100480, 113020427520, 1695306412800, 28432576972800, 483353808537600, 9056055981772800, 172065063653683200, 3562946373482496000, 74821873843132416000, 1697172166720622592000

Offset: 0

Author

Michael Somos, Mar 04 2004

Keywords

• nonn

Comments

Stirling transform of -(-1)^n*a(n-1)=[1,0,1,-3,18,...] is A052856(n-2)=[1,1,2,4,14,76,...].

Number of cycles of even cardinality in all permutations of [n]. Example: a(3)=3 because among (1)(2)(3), (1)(23), (12)(3), (13)(2), (132), (123) we have three cycles of even length. - Emeric Deutsch, Aug 12 2004

Examples

			a(4)=4!*(1/2+1/4)=18, a(5)=5!*(1/2+1/4)=90.
		

References

• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, Exercise 3.3.13.

Links

• N. J. A. Sloane and T. D. Noe, Table of n, a(n) for n = 0..200

Crossrefs

A046674(n)=a(2n). Cf. A081358, A151883, A151884.

Programs

Mathematica

nn = 20; Range[0, nn]! CoefficientList[
  D[Series[(1 - x^2)^(-y/2) ((1 + x)/(1 - x))^(1/2), {x, 0, nn}], y] /. y -> 1, x]  (* Geoffrey Critzer, Aug 27 2012 *)

PARI

a(n)=if(n<0,0,n!*sum(k=1,n\2,1/k)/2)

PARI

{a(n)=if(n<0, 0, n!*polcoeff( log(1-x^2+x*O(x^n))/(2*x-2), n))}

Formula

a(2n+1) = (2n+1)*a(2n).

From Vladeta Jovovic, Mar 06 2004: (Start)

a(n) = n!*(Psi(floor(n/2)+1)+gamma)/2.

E.g.f.: log(1-x^2)/(2*x-2). (End)

a(n) = n!/2*h(floor(n/2)), where h(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Jul 19 2011