A145886 - OEIS

%I #19 Jul 26 2022 15:43:30

%S 0,1,0,12,50,405,3234,29680,300348,3337425,40382540,528644556,

%T 7445076990,112248853717,1803999433950,30788257007040,556112892188504,

%U 10598857474652865,212565974908314168,4475073155964510700

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

%H Vincenzo Librandi, <a href="/A145886/b145886.txt">Table of n, a(n) for n = 1..300</a>

%H R. Mantaci and F. Rakotondrajao, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00531-6">Exceedingly deranging!</a>, Advances in Appl. Math., 30 (2003), 177-188.

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

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

%F a(n) ~ n!*exp(-1)*n/4. - _Vaclav Kotesovec_, Oct 07 2013

%F 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

%e 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.

%p 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);

%t 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 *)

%Y Cf. A145880, A145881, A145887.

%K nonn

%O 1,4

%A _Emeric Deutsch_, Nov 06 2008