A145886 Number of excedances in all odd permutations of {1,2,...,n} with no fixed points.

0, 1, 0, 12, 50, 405, 3234, 29680, 300348, 3337425, 40382540, 528644556, 7445076990, 112248853717, 1803999433950, 30788257007040, 556112892188504, 10598857474652865, 212565974908314168, 4475073155964510700

Offset

1,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..300

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

Formula

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

a(n) = Sum_{k=1..n-1} k * A145880(n,k), n>=2.

a(n) ~ n!*exp(-1)*n/4. - Vaclav Kotesovec, Oct 07 2013

D-finite with recurrence +(-3*n+7)*a(n) +(3*n+2)*(n-3)*a(n-1) +(3*n^2-n+16)*a(n-2) +(3*n^2-23*n+32)*a(n-3) +(3*n-5)*(n-3)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

Example

a(4)=12 because the odd derangements of {1,2,3,4} are 4123, 3142, 4312, 2413, 2341 and 3421, having 1, 2, 2, 2, 3 and 2, excedances, respectively.

Maple

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

Mathematica

Rest[CoefficientList[Series[1/4*x^2*(2-2*x+x^2)*E^(-x)/(1-x)^2, {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 07 2013 *)

Crossrefs

Cf. A145880, A145881, A145887.

Sequence in context: A051797 A342482 A333276 * A349860 A268351 A374384

Adjacent sequences: A145883 A145884 A145885 * A145887 A145888 A145889

Keyword

nonn

Author

Emeric Deutsch, Nov 06 2008

Status

approved