%I #21 Mar 19 2019 12:22:45
%S 0,0,0,0,6,30,180,1260,8820,79380,793800,8731800,106029000,1378377000,
%T 19297278000,289459170000,4627941318000,78675002406000,
%U 1416150043308000,26906850822852000,538156815464268000
%N Number of permutations of an n-set containing a 4-cycle.
%H Robert Israel, <a href="/A029571/b029571.txt">Table of n, a(n) for n = 0..449</a>
%F a(n) = n! * (1 - sum(k=0..floor(n/4), (-1)^k/(k!*4^k) ) ).
%F a(n)/n! is asymptotic to 1-e^(-1/4) = 1 - A092616.
%F a(n) = n! (1 - Gamma(floor(n/4)+1,-1/4)*exp(1/4)/(floor(n/4))!). - _Robert Israel_, Dec 07 2016
%F E.g.f.: (1-exp(-x^4/4))/(1-x). - _Alois P. Heinz_, Oct 11 2017
%p L:= [seq( 1 - add((-1)^k/(k!*4^k),k=0..m),m=0..10)]:
%p seq(seq((4*m+j)!*L[m+1],j=0..3),m=0..10); # _Robert Israel_, Dec 07 2016
%t a[n_] := n! (1 - Sum[(-1)^k/(k! 4^k), {k, 0, Floor[n/4]}]);
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 19 2019 *)
%o (PARI) a(n)=n! * (1 - sum(k=0,floor(n/4), (-1)^k/(k!*4^k) ) ); \\ _Joerg Arndt_, Aug 08 2013
%Y Column k=4 of A293211.
%K nonn
%O 0,5
%A _Rob Pratt_