A248087 Number of n-derangements that have an odd number of 2-cycles.

0, 0, 1, 0, 0, 20, 105, 504, 4480, 43560, 424305, 4613840, 55668096, 724667580, 10136511385, 152029000200, 2432747715840, 41357024915024, 744416488494945, 14143911946532640, 282878618744592640, 5940450667217358180, 130689899053015493961, 3005867708207562586520

Offset: 0

Author

Emeric Deutsch, Oct 15 2014

Keywords

• nonn

Examples

			a(2) = 1 because the 2-derangements in cycle notation are: (12).
a(3) = 0 because the 3-derangements in cycle notation are: (123),(132).
a(4) = 0 because the 4-derangements in cycle notation are: (1234),(1243),(1324),(1342),(1423),(1432),(12)(34),(13)(24),(14)(23).
		

References

• M. Bona, Combinatorics of Permutations. 2nd ed., Chapman and Hall/CRC Press, 2012, Boca Raton, FL. p. 123, Example 3.65.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A000166, A347106.

Programs

Maple

G := sinh((1/2)*z^2)*exp(-ln(1-z)-z-(1/2)*z^2): Gser := series(G, z = 0, 30): seq(factorial(n)*coeff(Gser, z, n), n = 0 .. 27);
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j,
     `if`(j=2, 1-t, t))*binomial(n-1, j-1)*(j-1)!, j=2..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Jan 27 2022

Mathematica

Rest[CoefficientList[Series[Sinh[x^2/2]/(E^(x*(2+x)/2)*(1-x)), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Oct 15 2014 *)

Formula

E.g.f.: sinh(z^2/2)*exp(-log(1-z)-z-z^2/2).

a(n) ~ n! * (exp(1)-1)/(2*exp(2)). - Vaclav Kotesovec, Oct 15 2014

From Benedict W. J. Irwin, May 24 2016: (Start)

Let y(-1)=0, y(0)=0, y(1)=1,

Let -2y(n)+y(n+1)-(n+1)y(n+2)+(n+2)y(n+3)=0,

a(n)=((-1)^n*2F0(1,-n;;1) - n!y(n+1))/2.

(End)