OFFSET
0,3
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.
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)
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 *)
PROG
(PARI) {a(n) = if(n<0, 0, if(n%2, 0, n!))}; /* Michael Somos, Mar 04 2004 */
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.
KEYWORD
nonn
AUTHOR
STATUS
approved