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

0,4

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

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

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

Example

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

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 *)

Prog

(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))}

Crossrefs

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

Sequence in context: A088336 A133594 A272492 * A064671 A363647 A058409

Adjacent sequences: A092688 A092689 A092690 * A092692 A092693 A092694

Keyword

nonn

Author

Michael Somos, Mar 04 2004

Status

approved