A005359 a(n) = n! if n is even, otherwise 0 (from Taylor series for cos x).

1, 0, 2, 0, 24, 0, 720, 0, 40320, 0, 3628800, 0, 479001600, 0, 87178291200, 0, 20922789888000, 0, 6402373705728000, 0, 2432902008176640000, 0, 1124000727777607680000, 0, 620448401733239439360000, 0

Offset: 0

Author

N. J. A. Sloane, Russ Cox

Keywords

• nonn

Comments

Normally sequences like this are not included, since with the alternating 0's deleted it is already in the database.

Stirling transform of a(n)=[0,2,0,24,0,720,...] is A052841(n)=[0,2,6,38,270,...]. - Michael Somos, Mar 04 2004

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

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

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

Also n-th derivative of arctanh(x) at x=0. - Michel Lagneau, Aug 13 2012

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

References

• Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought".

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Michael Somos, Number of permutations with all cycles of even length, answer to Mathematics Stack Exchange question 3152701, Mar 18 2019.
• Index entries for sequences related to factorial numbers

Crossrefs

From Johannes W. Meijer, Nov 12 2009: (Start)

Equals the first right hand column of A167565.

Equals the first left hand column of A167568.

(End)

Cf. A177145.

Bisection (even part) gives A010050.

Programs

Maple

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
a:=n->n!+(-1)^n*n!: seq(a(n)/2, n=0..25); # Zerinvary Lajos, Mar 25 2008

Mathematica

Riffle[Range[0,30,2]!,0] (* Harvey P. Dale, Nov 16 2011 *)
a[ n_] := If[n >= 0 && EvenQ[n], n!, 0]; (* Michael Somos, Mar 19 2019 *)

PARI

{a(n) = if(n<0, 0, if(n%2, 0, n!))}; /* Michael Somos, Mar 04 2004 */

Formula

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

a(n) = Product_{k=0..n/2-1} binomial(n-2k,2)*2^(n/2) for even n. - Geoffrey Critzer, Jun 05 2016

From Ilya Gutkovskiy, Jun 05 2016: (Start)

D-finite with recurrence a(n) = n*(n - 1)*a(n-2), a(0)=1, a(1)=0.

a(n) = n!*((-1)^n + 1)/2. (End)