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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

REFERENCES

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

LINKS

Table of n, a(n) for n=1..22.

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)

EXAMPLE

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).

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 = 1 .. 27);

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 *)

CROSSREFS

Cf. A000166.

Sequence in context: A063785 A181703 A187756 * A209547 A278642 A135174

Adjacent sequences:  A248084 A248085 A248086 * A248088 A248089 A248090

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 15 2014

STATUS

approved

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Last modified April 5 09:46 EDT 2020. Contains 333239 sequences. (Running on oeis4.)