A001465 Number of degree-n odd permutations of order 2. (Formerly M2538 N1003)

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

0,4

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).

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

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

Cf. A000085, A000704.

Sequence in context: A109490 A130760 A154134 * A094276 A151376 A066245

Adjacent sequences: A001462 A001463 A001464 * A001466 A001467 A001468

Keyword

nonn

Author

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

Extensions

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

Status

approved