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 A092691 a(n) = n! * Sum_{k=1..floor(n/2)} 1/(2k). 8
 0, 0, 1, 3, 18, 90, 660, 4620, 42000, 378000, 4142880, 45571680, 586776960, 7628100480, 113020427520, 1695306412800, 28432576972800, 483353808537600, 9056055981772800, 172065063653683200, 3562946373482496000, 74821873843132416000, 1697172166720622592000 (list; graph; refs; listen; history; text; internal format)
 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 A058409 A125833 Adjacent sequences:  A092688 A092689 A092690 * A092692 A092693 A092694 KEYWORD nonn AUTHOR Michael Somos, Mar 04 2004 STATUS approved

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Last modified February 16 18:53 EST 2019. Contains 320165 sequences. (Running on oeis4.)