A216778 - OEIS

%I #24 Nov 14 2022 20:03:00

%S 1,0,0,0,3,20,130,924,7413,66744,667476,7342280,88107415,1145396460,

%T 16035550518,240533257860,3848532125865,65425046139824,

%U 1177650830516968,22375365779822544,447507315596451051,9397653627525472260,206748379805560389930,4755212735527888968620

%N Number of derangements on n elements with an even number of cycles.

%H Michael De Vlieger, <a href="/A216778/b216778.txt">Table of n, a(n) for n = 0..450</a>

%H Paulo H. L. Martins, Ronald Dickman, and Robert M. Ziff, <a href="https://arxiv.org/abs/2211.04622">Percolation in two-species antagonistic random sequential adsorption in two dimensions</a>, arXiv:2211.04622 [cond-mat.stat-mech], 2022.

%F a(n+1) = n*(a(n) + a(n-1) + n - 2), a(0)=1, a(1)=0.

%F a(n) = (A000166(n) - n + 1)/2.

%F E.g.f.: cosh(log(1/(1-x)) - x). - _Geoffrey Critzer_, Jun 23 2014

%p a := proc (n) local x, y, t, k; if n = 0 then 1 elif n = 1 then 0 else x := 1; y := 0; for k from 2 to n do t := y; y := (k-1)*(x+y+k-3); x := t end do; y end if end proc;

%t nn=23;Range[0,nn]!*CoefficientList[Series[Cosh[Log[1/(1-x)]-x],{x,0,nn}],x] (* _Geoffrey Critzer_, Jun 23 2014 *)

%Y Cf. A000166, A216779 (derangements with odd number of cycles).

%K nonn,easy

%O 0,5

%A _José H. Nieto S._, Sep 16 2012