A145887 Number of excedances in all even permutations of {1,2,...,n} with no fixed points.

0, 0, 3, 6, 60, 390, 3255, 29652, 300384, 3337380, 40382595, 528644490, 7445077068, 112248853626, 1803999434055, 30788257006920, 556112892188640, 10598857474652712, 212565974908314339, 4475073155964510510

Offset

1,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188.

Formula

a(n) = Sum_{k=1..n-1} k*A145881(n,k), for n>=2.

E.g.f.: (1/4)*z^3*(2-z)*exp(-z)/(1-z)^2.

a(n) = 1/4*n*n!*Sum_{k=2..n-1} (-1)^k*(k+2)*(k-1)/(k+1)!. - Vaclav Kotesovec, Oct 28 2012

a(n) ~ n * n! / (4*exp(1)). - Vaclav Kotesovec, Dec 10 2021

D-finite with recurrence (-n+3)*a(n) +(n^2-3*n-2)*a(n-1) +(n-1)*(n+1)*a(n-2)=0. - R. J. Mathar, Jul 26 2022

Example

a(4)=6 because the even derangements of {1,2,3,4} are 3412, 2143 and 4321, having 2, 2 and 2, excedances, respectively.

Maple

G:=(1/4)*z^3*(2-z)*exp(-z)/(1-z)^2: Gser:=series(G, z=0, 30): seq(factorial(n)*coeff(Gser, z, n), n=1..21);

Mathematica

Table[1/4*n*n!*Sum[(-1)^k*(k+2)*(k-1)/(k+1)!, {k, 2, n-1}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 28 2012 *)

Crossrefs

Cf. A145880, A145881, A145886.

Sequence in context: A058161 A012877 A103066 * A364852 A219809 A137123

Adjacent sequences: A145884 A145885 A145886 * A145888 A145889 A145890

Keyword

nonn

Author

Emeric Deutsch, Nov 07 2008

Status

approved