%I A000757 M4521 N1915
%S A000757 1,0,0,1,1,8,36,229,1625,13208,120288,1214673,13469897,162744944,
%T A000757 2128047988,29943053061,451123462673,7245940789072,123604151490592,
%U A000757 2231697509543361,42519034050101745,852495597142800376
%N A000757 Number of cyclic permutations of [n] with no i->i+1 (mod n)
%D A000757 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000757 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000757 S. M. Jacob, The enumeration of the Latin rectangle of depth three..., 
               Proc. London Math. Soc., 31 (1928), 329-336.
%D A000757 R. P. Stanley, Space Programs Summary. Jet Propulsion Laboratory, California 
               Institute of Technology, Pasadena, California, Vol. 37-40-4 (1966), 
               pp. 208-214.
%D A000757 R. P. Stanley, Enumerative Combinatorics I, Chap. 2, Exercise 8, p. 88.
%D A000757 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, 
               Boca Raton, Florida, 2002, p. 182 and p. 183, Table 5.6.
%F A000757 a(n)=(-1)^n + sum((-1)^k*binomial(n, k)*(n-k-1)!, k=0..n-1); e.g.f.: 
               (1-ln(1-x))/e^x; a(n) = (n-3)*a(n-1) + (n-2)*(2*a(n-2) + a(n-3)).
%F A000757 a(n)=(n-2)*a(n-1)+(n-1)*a(n-2)-(-1)^n, n>0.
%F A000757 a(n)=sum(((-1)^(n-j))*D(j-1),j=3..n), n>=3, with the derangements numbers 
               (subfactorials) D(n)=A000166(n).
%e A000757 a(4)=1 because from the 4!/4=6 circular permutations of n=4 elements 
               (1,2,3,4), (1,4,3,2), (1,3,4,2),(1,2,4,3), (1,4,2,3) and (1,3,2,4) 
               only (1,4,3,2) has no successor pair (i,i+1). Note that (4,1) is 
               also a successor pair. W. Lang, Jan 21 2008.
%o A000757 (PARI) a(n)=if(n<0,0,(-1)^n + sum(k=0,n-1,(-1)^k*binomial(n,k)*(n-k-1)!))
%Y A000757 A000757(n)=(-1)^n+A002741(n).
%Y A000757 Sequence in context: A001555 A032770 A032794 this_sequence A126756 A058823 
               A050536
%Y A000757 Adjacent sequences: A000754 A000755 A000756 this_sequence A000758 A000759 
               A000760
%K A000757 nonn,easy,nice
%O A000757 0,6
%A A000757 N. J. A. Sloane (njas(AT)research.att.com).
%E A000757 Better description from Len Smiley (smiley(AT)math.uaa.alaska.edu)
%E A000757 Additional comments from Michael Somos, Jun 21, 2002