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