%I M2538 N1003 #55 Oct 24 2020 04:28:10

%S 0,0,1,3,6,10,30,126,448,1296,4140,17380,76296,296088,1126216,4940040,

%T 23904000,110455936,489602448,2313783216,11960299360,61878663840,

%U 309644323296,1587272962528,8699800221696,48793502304000,268603261201600,1487663739072576

%N Number of degree-n odd permutations of order 2.

%C Number of even partitions of an n-element set avoiding the pattern 123 (see Goyt paper). - _Ralf Stephan_, May 08 2007

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A001465/b001465.txt">Table of n, a(n) for n = 0..800</a>

%H Lev Glebsky, Melany Licón, and Luis Manuel Rivera, <a href="https://arxiv.org/abs/1907.00548">On the number of even roots of permutations</a>, arXiv:1907.00548 [math.CO], 2019.

%H A. M. Goyt, <a href="https://arxiv.org/abs/math/0603481">Avoidance of partitions of a 3-element set</a>, arXiv:math/0603481 [math.CO], 2006-2007.

%H L. Moser and M. Wyman, <a href="http://dx.doi.org/10.4153/CJM-1955-020-0">On solutions of x^d = 1 in symmetric groups</a>, Canad. J. Math., 7 (1955), 159-168.

%F a(n) = Sum_{i=0..floor((n-2)/4)} C(n,4i+2)*(4i+2)!/(4i+2)!!. - _Ralf Stephan_, May 08 2007

%F 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

%F From _Jianing Song_, Oct 24 2020: (Start)

%F E.g.f.: exp(x)*sinh(x^2/2).

%F a(n) = A000085(n) - A000704(n). (End)

%e 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

%p a:= proc(n) option remember; `if`(n<4, (n-1)*n/2,

%p ((2*n-3)*a(n-1)-(n-1)*a(n-2))/(n-2)+(n-1)*(n-3)*a(n-4))

%p end:

%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 24 2018

%t 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 *)

%Y Cf. A000085, A000704.

%K nonn

%O 0,4

%A _N. J. A. Sloane_ and _J. H. Conway_

%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 11 2004