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%I #51 Aug 12 2022 19:59:28
%S 1,0,2,0,24,0,720,0,40320,0,3628800,0,479001600,0,87178291200,0,
%T 20922789888000,0,6402373705728000,0,2432902008176640000,0,
%U 1124000727777607680000,0,620448401733239439360000,0
%N a(n) = n! if n is even, otherwise 0 (from Taylor series for cos x).
%C Normally sequences like this are not included, since with the alternating 0's deleted it is already in the database.
%C Stirling transform of a(n)=[0,2,0,24,0,720,...] is A052841(n)=[0,2,6,38,270,...]. - _Michael Somos_, Mar 04 2004
%C Stirling transform of a(n-1)=[1,0,2,0,24,0,...] is A000670(n-1)=[1,1,3,13,75,...]. - _Michael Somos_, Mar 04 2004
%C Stirling transform of a(n-1)=[0,0,2,0,24,0,...] is A052875(n-1)=[0,0,2,12,74,...]. - _Michael Somos_, Mar 04 2004
%C Stirling transform of (-1)^n*A052811(n)=[0,2,-6,46,-340,...] is a(n)=[0,2,0,24,0,...]. - _Michael Somos_, Mar 04 2004
%C Also n-th derivative of arctanh(x) at x=0. - _Michel Lagneau_, Aug 13 2012
%C Binomial convolution square of A177145 (with offset 0) because each permutation in S_{2n} uniquely determines a bi-partition of its elements into even and odd cycles and these are both enumerated by A177145. - _Michael Somos_, Mar 19 2019
%D Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought".
%H Vincenzo Librandi, <a href="/A005359/b005359.txt">Table of n, a(n) for n = 0..200</a>
%H Michael Somos, <a href="https://math.stackexchange.com/q/3152701">Number of permutations with all cycles of even length</a>, answer to Mathematics Stack Exchange question 3152701, Mar 18 2019.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F E.g.f. 1/(1-x^2) = d/dx log(sqrt((1+x)/(1-x))). a(2n)=(2n)!, a(2n+1)=0. - _Michael Somos_, Mar 04 2004
%F a(n) = Product_{k=0..n/2-1} binomial(n-2k,2)*2^(n/2) for even n. - _Geoffrey Critzer_, Jun 05 2016
%F From _Ilya Gutkovskiy_, Jun 05 2016: (Start)
%F D-finite with recurrence a(n) = n*(n - 1)*a(n-2), a(0)=1, a(1)=0.
%F a(n) = n!*((-1)^n + 1)/2. (End)
%p BB:={E=Prod(Z,Z),S=Union(Epsilon,Prod(S,E))}: ZL:=[S,BB, labeled]: > seq(count(ZL,size=n),n=0..25); # _Zerinvary Lajos_, Apr 22 2007
%p a:=n->n!+(-1)^n*n!: seq(a(n)/2, n=0..25); # _Zerinvary Lajos_, Mar 25 2008
%t Riffle[Range[0,30,2]!,0] (* _Harvey P. Dale_, Nov 16 2011 *)
%t a[ n_] := If[n >= 0 && EvenQ[n], n!, 0]; (* _Michael Somos_, Mar 19 2019 *)
%o (PARI) {a(n) = if(n<0, 0, if(n%2, 0, n!))}; /* _Michael Somos_, Mar 04 2004 */
%Y From _Johannes W. Meijer_, Nov 12 2009: (Start)
%Y Equals the first right hand column of A167565.
%Y Equals the first left hand column of A167568.
%Y (End)
%Y Cf. A177145.
%Y Bisection (even part) gives A010050.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, _Russ Cox_
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