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%I #24 Jul 13 2015 14:12:12
%S 0,0,1,3,18,90,660,4620,42000,378000,4142880,45571680,586776960,
%T 7628100480,113020427520,1695306412800,28432576972800,483353808537600,
%U 9056055981772800,172065063653683200,3562946373482496000,74821873843132416000,1697172166720622592000
%N a(n) = n! * Sum_{k=1..floor(n/2)} 1/(2k).
%C Stirling transform of -(-1)^n*a(n-1)=[1,0,1,-3,18,...] is A052856(n-2)=[1,1,2,4,14,76,...].
%C 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
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, Exercise 3.3.13.
%H N. J. A. Sloane and T. D. Noe, <a href="/A092691/b092691.txt">Table of n, a(n) for n = 0..200</a>
%F a(2n+1) = (2n+1)*a(2n).
%F From _Vladeta Jovovic_, Mar 06 2004: (Start)
%F a(n) = n!*(Psi(floor(n/2)+1)+gamma)/2.
%F E.g.f.: log(1-x^2)/(2*x-2). (End)
%F a(n) = n!/2*h(floor(n/2)), where h(n) = Sum_{k=1..n} 1/k. - _Gary Detlefs_, Jul 19 2011
%e a(4)=4!*(1/2+1/4)=18, a(5)=5!*(1/2+1/4)=90.
%t nn = 20; Range[0, nn]! CoefficientList[
%t 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 *)
%o (PARI) a(n)=if(n<0,0,n!*sum(k=1,n\2,1/k)/2)
%o (PARI) {a(n)=if(n<0, 0, n!*polcoeff( log(1-x^2+x*O(x^n))/(2*x-2), n))}
%Y A046674(n)=a(2n). Cf. A081358, A151883, A151884.
%K nonn
%O 0,4
%A _Michael Somos_, Mar 04 2004
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