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
Keywords
Examples
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.
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.
Programs
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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);
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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 *)
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