OFFSET
0,4
COMMENTS
Number of even partitions of an n-element set avoiding the pattern 123 (see Goyt paper). - Ralf Stephan, May 08 2007
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..800
Lev Glebsky, Melany Licón, and Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.
A. M. Goyt, Avoidance of partitions of a 3-element set, arXiv:math/0603481 [math.CO], 2006-2007.
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
FORMULA
a(n) = Sum_{i=0..floor((n-2)/4)} C(n,4i+2)*(4i+2)!/(4i+2)!!. - Ralf Stephan, May 08 2007
Conjecture: a(n) -3*a(n-1) +3*a(n-2) -a(n-3) -(n-1)*(n-3)*a(n-4) +(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, May 30 2014
From Jianing Song, Oct 24 2020: (Start)
E.g.f.: exp(x)*sinh(x^2/2).
EXAMPLE
For n=3, a(3)=3 and (1,2), (1, 3), (2, 3) are all the degree-2 odd permutations of order 2. - Luis Manuel Rivera Martínez, May 22 2018
MAPLE
a:= proc(n) option remember; `if`(n<4, (n-1)*n/2,
((2*n-3)*a(n-1)-(n-1)*a(n-2))/(n-2)+(n-1)*(n-3)*a(n-4))
end:
seq(a(n), n=0..30); # Alois P. Heinz, May 24 2018
MATHEMATICA
Table[Sum[Binomial[n , 4 i + 2] (4 i + 2)!/(2^(2 i + 1) (2 i + 1)!), {i, 0, Floor[(n - 2)/4]}], {n, 0, 22}] (* Luis Manuel Rivera Martínez, May 22 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 11 2004
STATUS
approved