A001465 Number of degree-n odd permutations of order 2.

0, 0, 1, 3, 6, 10, 30, 126, 448, 1296, 4140, 17380, 76296, 296088, 1126216, 4940040, 23904000, 110455936, 489602448, 2313783216, 11960299360, 61878663840, 309644323296, 1587272962528, 8699800221696, 48793502304000, 268603261201600, 1487663739072576

Offset: 0

Author

N. J. A. Sloane and J. H. Conway

Keywords

• nonn

Comments

Number of even partitions of an n-element set avoiding the pattern 123 (see Goyt paper). - Ralf Stephan, May 08 2007

Examples

			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
		

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.

Crossrefs

Cf. A000085, A000704.

Programs

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 *)

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).

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

Extensions

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