# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a000757 Showing 1-1 of 1 %I A000757 M4521 N1915 #74 Aug 30 2023 07:28:18 %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 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, pp. 182, 183, Table 5.6. %D A000757 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000757 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %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 N. Ya. Vilenkin, Combinatorics, pp. 56-57, Q_n = a(n), n >= 3. Academic Press, 1971. On the Merry Go-Round. %H A000757 Vincenzo Librandi, Table of n, a(n) for n = 0..200 %H A000757 Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - _N. J. A. Sloane_, Feb 06 2013 %H A000757 Bhadrachalam Chitturi and Krishnaveni K S, Adjacencies in Permutations, arXiv preprint arXiv:1601.04469 [cs.DM], 2016. %H A000757 S. M. Jacob, The enumeration of the Latin rectangle of depth three..., Proc. London Math. Soc., 31 (1928), 329-336. %H A000757 R. Moreno, L. M. Rivera, Blocks in Cycles and k-commuting Permutations, arXiv preprint arXiv:1306.5708 [math.CO], 2013. %H A000757 Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014. %H A000757 R. P. Stanley, Permutations with no runs of length 2, Space Programs Summary. Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, Vol. 37-40-4 (1966), pp. 208-214. [Annotated scanned copy] %F A000757 From _Michael Somos_, Jun 21 2002: (Start) %F A000757 a(n) = (-1)^n + Sum_{k=0..n-1} (-1)^k*binomial(n, k)*(n-k-1)!; %F A000757 e.g.f.: (1 - log(1 - x)) / e^x; %F A000757 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, if n > 0. %F A000757 a(n) = (-1)^n + A002741(n). (End) %F A000757 a(n) = n-th forward difference of [1, 1, 1, 2, 6, 24, ...] (factorials A000142 with 1 prepended). - _Michael Somos_, Mar 28 2011 %F A000757 a(n) = Sum_{j=3..n} (-1)^(n-j)*D(j-1), n >= 3, with the derangements numbers (subfactorials) D(n) = A000166(n). %F A000757 a(n) + a(n+1) = A000166(n). - _Aaron Meyerowitz_, Feb 08 2014 %F A000757 a(n) ~ exp(-1)*(n-1)! * (1 - 1/n + 1/n^3 + 1/n^4 - 2/n^5 - 9/n^6 - 9/n^7 + 50/n^8 + 267/n^9 + 413/n^10 + ...), numerators are A000587. - _Vaclav Kotesovec_, Jul 03 2016 %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. - _Wolfdieter Lang_, Jan 21 2008 %e A000757 a(3) = 1 = 2! - 3*1! + 3*0! - 1. a(4) = 1 = 3! - 4*2! + 6*1! - 4*0! + 1. - _Michael Somos_, Mar 28 2011 %e A000757 G.f. = 1 + x^3 + x^4 + 8*x^5 + 36*x^6 + 229*x^7 + 1625*x^8 + 13208*x^9 + ... %t A000757 a[n_] := (-1)^n + Sum[(-1)^k*n!/((n-k)*k!), {k, 0, n-1}]; Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Aug 30 2011, after _Michael Somos_ *) %o A000757 (PARI) {a(n) = if( n<0, 0, (-1)^n + sum( k=0, n-1, (-1)^k * binomial( n, k) * (n - k - 1)!))}; /* _Michael Somos_, Jun 21 2002 */ %Y A000757 Cf. A000142, A002741. %K A000757 nonn,easy,nice %O A000757 0,6 %A A000757 _N. J. A. Sloane_ %E A000757 Better description from _Len Smiley_ %E A000757 Additional comments from _Michael Somos_, Jun 21 2002 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE