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A000166 Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.
(Formerly M1937 N0766)
1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 2355301661033953, 44750731559645106, 895014631192902121, 18795307255050944540, 413496759611120779881, 9510425471055777937262 (list; graph; refs; listen; history; text; internal format)



Euler not only gives the first ten or so terms of the sequence, he also proves both recurrences a(n) = (n-1)(a(n-1) + a(n-2)) and a(n) = n*a(n-1) + (-1)^n.

a(n) is the permanent of the matrix with 0 on the diagonal and 1 elsewhere. - Yuval Dekel, Nov 01 2003

a(n) is the number of desarrangements of length n. A desarrangement of length n is a permutation p of {1,2,...,n} for which the smallest of all the ascents of p (taken to be n if there are no ascents) is even. Example: a(3) = 2 because we have 213 and 312 (smallest ascents at i = 2). See the J. Désarménien link and the Bona reference (p. 118). - Emeric Deutsch, Dec 28 2007

a(n) is the number of deco polyominoes of height n and having in the last column an even number of cells. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. - Emeric Deutsch, Dec 28 2007

Attributed to Nicholas Bernoulli in connection with a probability problem that he presented. See Problem #15, p. 494, in "History of Mathematics" by David M. Burton, 6th edition. - Mohammad K. Azarian, Feb 25 2008

a(n) is the number of permutations p of {1,2,...,n} with p(1)!=1 and having no right-to-left minima in consecutive positions. Example a(3) = 2 because we have 231 and 321. - Emeric Deutsch, Mar 12 2008

a(n) is the number of permutations p of {1,2,...,n} with p(n)! = n and having no left to right maxima in consecutive positions. Example a(3) = 2 because we have 312 and 321. - Emeric Deutsch, Mar 12 2008

Number of wedged (n-1)-spheres in the homotopy type of the Boolean complex of the complete graph K_n. - Bridget Tenner, Jun 04 2008

The only prime number in the sequence is 2. - Howard Berman (howard_berman(AT)hotmail.com), Nov 08 2008

From Emeric Deutsch, Apr 02 2009: (Start)

a(n) is the number of permutations of {1,2,...,n} having exactly one small ascent. A small ascent in a permutation (p_1,p_2,...,p_n) is a position i such that p_{i+1} - p_i = 1. (Example: a(3) = 2 because we have 312 and 231; see the Charalambides reference, pp. 176-180.) [See also David, Kendall and Barton, p. 263. - N. J. A. Sloane, Apr 11 2014]

a(n) is the number of permutations of {1,2,...,n} having exactly one small descent. A small descent in a permutation (p_1,p_2,...,p_n) is a position i such that p_i - p_{i+1} = 1. (Example: a(3)=2 because we have 132 and 213.) (End)

For n>2, a(n) + a(n-1) = A000255(n-1); where A000255 = (1, 1, 3, 11, 53, ...). - Gary W. Adamson, Apr 16 2009

Connection to A002469 (game of mousetrap with n cards): A002469(n) = (n-2)*A000255(n-1) + A000166(n). (Cf. triangle A159610). - Gary W. Adamson, Apr 17 2009

From Emeric Deutsch, Jul 18 2009: (Start)

a(n) is the sum of the values of the largest fixed points of all non-derangements of length n-1. Example: a(4)=9 because the non-derangements of length 3 are 123, 132, 213, and 321, having largest fixed points 3, 1, 3, and 2, respectively.

a(n) is the number of non-derangements of length n+1 for which the difference between the largest and smallest fixed point is 2. Example: a(3) = 2 because we have 1'43'2 and 32'14'; a(4) = 9 because we have 1'23'54, 1'43'52, 1'53'24, 52'34'1, 52'14'3, 32'54'1, 213'45', 243'15', and 413'25' (the extreme fixed points are marked).


a(n), n>=1, is also the number of unordered necklaces with n beads, labeled differently from 1 to n, where each necklace has >=2 beads. This produces the M2 multinomial formula involving partitions without part 1 given below. Because M2(p) counts the permutations with cycle structure given by partition p, this formula gives the number of permutations without fixed points (no 1-cycles), i.e., the derangements, hence the subfactorials with their recurrence relation and inputs. Each necklace with no beads is assumed to contribute a factor 1 in the counting, hence a(0)=1. This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010). - Wolfdieter Lang, Jun 01 2010

From Emeric Deutsch, Sep 06 2010: (Start)

a(n) is the number of permutations of {1,2,...,n, n+1} starting with 1 and having no successions. A succession in a permutation (p_1,p_2,...,p_n) is a position i such that p_{i+1} - p_i = 1. Example: a(3)=2 because we have 1324 and 1432.

a(n) is the number of permutations of {1,2,...,n} that do not start with 1 and have no successions. A succession in a permutation (p_1,p_2,...,p_n) is a position i such that p_{i+1} - p_i = 1. Example: a(3)=2 because we have 213 and 321.


Increasing colored 1-2 trees with choice of two colors for the rightmost branch of nonleave except on the leftmost path, there is no vertex of outdegree one on the leftmost path. - Wenjin Woan, May 23 2011

a(n) = number of zeros in n-th row of the triangle in A170942, n > 0. - Reinhard Zumkeller, Mar 29 2012

a(n) is the maximal number of totally mixed Nash equilibria in games of n players, each with 2 pure options. - Raimundas Vidunas, Jan 22 2014

Convolution of sequence A135799 with the sequence generated by 1+x^2/(2*x+1). - Thomas Baruchel, Jan 08 2016

The number of interior lattice points of the subpolytope of the n-dimensional permutohedron whose vertices correspond to permutations avoiding 132 and 312. - Robert Davis, Oct 05 2016

Consider n circles of different radii, where each circle is either put inside some bigger circle or contains a smaller circle inside it (no common points are allowed). Then a(n) gives the number of such combinations. - Anton Zakharov, Oct 12 2016

If we partition the permutations of [n+1] in A000240 according to their starting digit, we will get (n+1) equinumerous classes each of size a(n), i.e., A000240(n+1) = (n+1)*a(n), hence a(n) is the size of each class of permutations of [n+1] in A000240. For example, for n = 4 we have 45 = 5*9. - Enrique Navarrete, Jan 10 2017

Call d_n1 the permutations of [n] that have the substring n1 but no substring in {12,23,...,(n-1)n}. If we partition them according to their starting digit, we will get (n-1) equinumerous classes each of size A000166(n-2) (the class starting with the digit 1 is empty since we must have the substring n1). Hence d_n1 = (n-1)*A000166(n-2) and A000166(n-2) is the size of each nonempty class in d_n1. For example, d_71 = 6*44 = 264, so there are 264 permutations in d_71 distributed in 6 nonempty classes of size A000166(5) = 44. (To get permutations in d_n1 recursively from more basic ones see the link "Forbidden Patterns" below.) - Enrique Navarrete, Jan 15 2017

Also the number of maximum matchings and minimum edge covers in the n-crown graph. - Eric W. Weisstein, Jun 14 and Dec 24 2017

The sequence a(n) taken modulo a positive integer k is periodic with exact period dividing k when k is even and dividing 2*k when k is odd. This follows from the congruence a(n+k) = (-1)^k*a(n) (mod k) for all n and k, which in turn is easily proved by induction making use of the recurrence a(n) = n*a(n-1) + (-1)^n. - Peter Bala, Nov 21 2017

a(n) is the number of unique possible solutions for a directed, no self loop containing graph, that has n vertices, and each vertice has an in- and out-degree of exactly 1. - Patrik Holopainen, Sep 18 2018


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P. R. de Montmort, On the Game of Thirteen (1713), reprinted in Annotated Readings in the History of Statistics, ed. H. A. David and A. W. F. Edwards, Springer-Verlag, 2001, pp. 25-29.

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H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 19.

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J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.

A. Hald, A History of Probability and Statistics and Their Applications Before 1750, Wiley, NY, 1990 (Chapter 19).

Kaufmann, Arnold. "Introduction à la combinatorique en vue des applications." Dunod, Paris, 1968. See p. 92.

Florian Kerschbaum and Orestis Terzidis, Filtering for Private Collaborative Benchmarking, in Emerging Trends in Information and Communication Security, Lecture Notes in Computer Science, Volume 3995/2006,

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P. A. MacMahon, Combinatory Analysis, 2 vols., Chelsea, NY, 1960, see p. 102.

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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 147, Eq. 5.2.9 (q=1).


Seiichi Manyama, Table of n, a(n) for n = 0..450 (terms 0..200 from T. D. Noe)

Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443.

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Joerg Arndt, Matters Computational (The Fxtbook), p. 280

Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7.

Roland Bacher, P. De La Harpe, Conjugacy growth series of some infinitely generated groups, arXiv:1603.07943 [math.GR], 2016.

B. Balof, H. Jenne, Tilings, Continued Fractions, Derangements, Scramblings, and e, Journal of Integer Sequences, 17 (2014), #14.2.7.

V. Baltic, On the number of certain types of strongly restricted permutations, Appl. An. Disc. Math. 4 (2010), 119-135; Doi:10.2298/AADM1000008B.

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P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D14 (1976), 1536-1553.

S. K. Das and N. Deo, Rencontres graphs: a family of bipartite graphs, Fib. Quart., Vol. 25, No. 3, August 1987, 250-262.

R. Davis and B. Sagan, Pattern-Avoiding Polytopes, arXiv preprint arxiv:1609.01782 [math.CO], 2016.

J. Désarménien, Une autre interpretation des nombres de derangements, Sem. Loth. Comb. B08b (1982) 11-16.

E. Deutsch and S. Elizalde, The largest and the smallest fixed points of permutations, arXiv:0904.2792 [math.CO], 2009.

R. M. Dickau, Derangements

Tomislav Došlic, Darko Veljan, Logarithmic behavior of some combinatorial sequences, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019)

P. Duchon, R. Duvignau, A new generation tree for permutations, FPSAC 2014, Chicago, USA; Discrete Mathematics and Theoretical Computer Science (DMTCS) Proceedings, 2014, 679-690.

J. East, R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

Uriel Feige, Tighter bounds for online bipartite matching, 2018.

Philip Feinsilver and John McSorley, Zeons, Permanents, the Johnson Scheme, and Generalized Derangements, International Journal of Combinatorics, Volume 2011, Article ID 539030, 29 pages.

FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation

H. Fripertinger, The Rencontre Numbers

Hannah Fry and Brady Haran, The Problems with Secret Santa, Numberphile video (2016)

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Mehdi Hassani, Derangements and Applications , Journal of Integer Sequences, Vol. 6 (2003), #03.1.2

M. Hassani, Counting and computing by e, arXiv:math/0606613 [math.CO], 2006.

Nick Hobson, Python program

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E. Irurozki, B. Calvo, J. A. Lozano, Sampling and learning the Mallows and Weighted Mallows models under the Hamming distance, 2014.

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G. Villemin's Almanach of Numbers, Sous-factorielle

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Eric Weisstein's World of Mathematics, Crown Graph

Eric Weisstein's World of Mathematics, Derangement

Eric Weisstein's World of Mathematics, Edge Cover

Eric Weisstein's World of Mathematics, Exponential Distribution

Eric Weisstein's World of Mathematics, Matching

Eric Weisstein's World of Mathematics, Maximum Independent Edge Set

Eric Weisstein's World of Mathematics, Rooks Problem

Eric Weisstein's World of Mathematics, Subfactorial

Wikipedia, Derangement

Wikipedia, Rencontres numbers

H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 176, Eq. 5.2.9 (q=1).

E. M. Wright, Arithmetical properties of Euler's rencontre number, J. London Math. Soc., (2) (1971/1972), 437-442.

D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.

Index entries for "core" sequences

OEIS Wiki, Derangement numbers

OEIS Wiki, Rencontres numbers


(A000166 + A000522)/2 = A009179, (A000166 - A000522)/2 = A009628.

The termwise sum of this sequence and A003048 gives the factorial numbers. - D. G. Rogers, Aug 26 2006

a(n) = {(n-1)!/exp(1)}, n > 1, where {x} is the nearest integer function. - Simon Plouffe, March 1993 [This uses offset 1, see below for the version with offset 0. - Charles R Greathouse IV, Jan 25 2012]

a(0) = 1, a(n) = floor(n!/e + 1/2) for n > 0.

a(n) = n!*Sum_{k=0..n}(-1)^k/k!.

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

a(n) = n*a(n-1) + (-1)^n.

E.g.f.: exp(-x)/(1-x).

O.g.f. for number of permutations with exactly k fixed points is (1/k!)*Sum_{i>=k} i!*x^i/(1+x)^(i+1). - Vladeta Jovovic, Aug 12 2002

E.g.f. for number of permutations with exactly k fixed points is x^k/(k!*exp(x)*(1-x)). - Vladeta Jovovic, Aug 25 2002

a(n) = Sum_{k=0..n} binomial(n, k)(-1)^(n-k)k! = Sum_{k=0..n} (-1)^(n-k)*n!/(n-k)!}. - Paul Barry, Aug 26 2004

The e.g.f. y(x) satisfies y' = x*y/(1-x).

Inverse binomial transform of A000142. - Ross La Haye, Sep 21 2004

Subf(n) = n^(n-1) - ( 2*C(n-2, 0) + 2*C(n-2, 1) + C(n-2, 2) )*n^(n-2) + ( 4*C(n-3, 0) + 11*C(n-3, 1) + 16*C(n-3, 2) + 11*C(n-3, 3) + 3*C(n-3, 4) )*n^(n-3) - ( 10*C(n-4, 0) + 55*C(n-4, 1) + 147*C(n-4, 2) + 215*C(n-4, 3) + 179*C(n-4, 4) + 80*C(n-4, 5) + 15*C(n-4, 6) )*n^(n-4) + ... . - André F. Labossière, Dec 06 2004

In Maple notation, representation as n-th moment of a positive function on [ -1, infinity]: a(n)= int( x^n*exp(-x-1), x=-1..infinity ), n=0, 1... . a(n) is the Hamburger moment of the function exp(-1-x)*Heaviside(x+1). - Karol A. Penson, Jan 21 2005

a(n) = A001120(n) - n!. - Philippe Deléham, Sep 04 2005

a(n) = Integral_{x=0..infinity} (x-1)^n*exp(-x) dx. - Gerald McGarvey, Oct 14 2006

a(n) = Sum_{k=2,4,...} T(n,k), where T(n,k) = A092582(n,k) = k*n!/(k+1)! for 1 <= k < n and T(n,n)=1. - Emeric Deutsch, Feb 23 2008

a(n) = n!/e + (-1)^n*(1/(n+2 - 1/(n+3 - 2/(n+4 - 3/(n+5 - ...))))). Asymptotic result (Ramanujan): (-1)^n*(a(n) - n!/e) ~ 1/n - 2/n^2 + 5/n^3 - 15/n^4 + ..., where the sequence [1,2,5,15,...] is the sequence of Bell numbers A000110. - Peter Bala, Jul 14 2008

From William Vaughn (wvaughn(AT)cvs.rochester.edu), Apr 13 2009: (Start)

a(n) = Integral_{p=0..1} (log(1/(1-p)) - 1)^n dp.

Proof: Using the substitutions 1=log(e) and y = e(1-p) the above integral can be converted to ((-1)^n/e) Integral_{y=0..e} (log(y))^n dy.

From CRC Integral tables we find the antiderivative of (log(y))^n is (-1)^n n! Sum_{k=0..n} (-1)^k y(log(y))^k / k!.

Using the fact that e(log(e))^r = e for any r>=0 and 0(log(0))^r = 0 for any r>=0 the integral becomes n! Sum_{k=0..n} (-1)^k / k!, which is line 9 of the Formula section. (End)

a(n) = exp(-1)*GAMMA(n+1,-1) (incomplete Gamma function). - Mark van Hoeij, Nov 11 2009

G.f.: 1/(1-x^2/(1-2x-4x^2/(1-4x-9x^2/(1-6x-16x^2/(1-8x-25x^2/(1-... (continued fraction). - Paul Barry, Nov 27 2009

a(n) = Sum_{p in Pano1(n)} M2(p), n>=1, with Pano1(n) the set of partitions without part 1, and the multinomial M2 numbers. See the characteristic array for partitions without part 1 given by A145573 in Abramowitz-Stegun (A-S) order, with A002865(n) the total number of such partitions. The M2 numbers are given for each partition in A-St order by the array A036039. - Wolfdieter Lang, Jun 01 2010

a(n) = row sum of A008306(n), n>1. - Gary Detlefs, Jul 14 2010

a(n) = ((-1)^n)*(n-1)*hypergeom([ -n+2, 2], [], 1), n>=1; 1 for n=0. - Wolfdieter Lang, Aug 16 2010

a(n) = (-1)^n * hypergeom([ -n, 1], [], 1), n>=1; 1 for n=0. From the binomial convolution due to the e.g.f. - Wolfdieter Lang, Aug 26 2010

Integral_{x=0..1} x^n*exp(x) = (-1)^n*(a(n)*e - n!).

O.g.f.: Sum_{n>=0} n^n*x^n/(1 + (n+1)*x)^(n+1). - Paul D. Hanna, Oct 06 2011

Abs((a(n) + a(n-1))*e - (A000142(n) + A000142(n-1)) < 2/n. - Seiichi Kirikami, Oct 17 2011

G.f.: hypergeom([1,1],[],x/(x+1))/(x+1). - Mark van Hoeij, Nov 07 2011

From Sergei N. Gladkovskii, Nov 25 2011, Jul 05 2012, Sep 23 2012, Oct 13 2012, Mar 09 2013, Mar 10 2013, Oct 18 2013 (Start) Continued fractions:

In general, e.g.f. (1+a*x)/exp(b*x) = U(0) with U(k) = 1+a*x/(1-b/(b-a*(k+1)/U(k+1))). For a=-1, b=-1: exp(-x)/(1-x) = 1/U(0).

E.g.f.: (1-x/(U(0)+x))/(1-x), where U(k) = k+1 - x + (k+1)*x/U(k+1).

E.g.f.: 1/Q(0) where Q(k) = 1 - x/(1 - 1/(1 - (k+1)/Q(k+1))).

G.f.: 1/U(0) where U(k) = 1 + x - x*(k+1)/(1 - x*(k+1)/U(k+1)).

G.f.: Q(0)/(1+x) where Q(k) = 1 + (2*k+1)*x/((1+x)-2*x*(1+x)*(k+1)/(2*x*(k+1)+(1+x)/ Q(k+1))).

G.f.: 1/Q(0) where Q(k) =  1 - 2*k*x - x^2*(k + 1)^2/Q(k+1).

G.f.: T(0) where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2-(1-2*x*k)*(1-2*x-2*x*k)/T(k+1)). (End)

0 = a(n)*(a(n+1) + a(n+2) - a(n+3)) + a(n+1)*(a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2)*a(n+2) if n>=0. - Michael Somos, Jan 25 2014

a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(k + x)^k*(k + x + 1)^(n-k) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(k + x)^(n-k)*(k + x - 1)^k, for arbitrary x. - Peter Bala, Feb 19 2017

From Peter Luschny, Jun 20 2017: (Start)

a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(-j-1, -n-1)*abs(Stirling1(j, k)).

a(n) = Sum_{k=0..n}(-1)^(n-k)*Pochhammer(n-k+1, k) (cf. A008279). (End)


a(2) = 1, a(3) = 2 and a(4) = 9 since the possibilities are {BA}, {BCA, CAB} and {BADC, BCDA, BDAC, CADB, CDAB, CDBA, DABC, DCAB, DCBA}. - Henry Bottomley, Jan 17 2001

The Boolean complex of the complete graph K_4 is homotopy equivalent to the wedge of 9 3-spheres.

Necklace problem for n = 6: partitions without part 1 and M2 numbers for n = 6: there are A002865(6) = 4 such partitions, namely (6), (2,4), (3^2) and (2^3) in A-St order with the M2 numbers 5!, 90, 40 and 15, respectively, adding up to 265 = a(6). This corresponds to 1 necklace with 6 beads, two necklaces with 2 and 4 beads respectively, two necklaces with 3 beads each and three necklaces with 2 beads each. - Wolfdieter Lang, Jun 01 2010

G.f. = 1 + x^2 + 9*x^3 + 44*x^4 + 265*x^5 + 1854*x^6 + 14833*x^7 + 133496*x^8 + ...


A000166 := proc(n) option remember; if n<=1 then 1-n else (n-1)*(procname(n-1)+procname(n-2)); fi; end;

a:=n->n!*sum((-1)^k/k!, k=0..n): seq(a(n), n=0..21); # Zerinvary Lajos, May 17 2007

ZL1:=[S, {S=Set(Cycle(Z, card>1))}, labeled]: seq(count(ZL1, size=n), n=0..21); # Zerinvary Lajos, Sep 26 2007

with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: A000166:=a(2):seq(count(A000166, size=n), n=0..21); # Zerinvary Lajos, Oct 02 2007

Z := (x, m)->m!^2*sum(x^j/((m-j)!^2), j=0..m): R := (x, n, m)->Z(x, m)^n: f := (t, n, m)->sum(coeff(R(x, n, m), x, j)*(t-1)^j*(n*m-j)!, j=0..n*m): seq(f(0, n, 1), n=0..21); # Zerinvary Lajos, Jan 22 2008

a:=proc(n) if `mod`(n, 2)=1 then sum(2*k*factorial(n)/factorial(2*k+1), k=1.. floor((1/2)*n)) else 1+sum(2*k*factorial(n)/factorial(2*k+1), k=1..floor((1/2)*n)-1) end if end proc: seq(a(n), n=0..20); # Emeric Deutsch, Feb 23 2008

G(x):=2*exp(-x)/(1-x): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n]/2, n=0..21); # Zerinvary Lajos, Apr 03 2009


a[0] = 1; a[n_] := n*a[n - 1] + (-1)^n; a /@ Range[0, 21] (* Robert G. Wilson v *)

a[0] = 1; a[1] = 0; a[n_] := Round[n!/E] /; n >= 1 (* Michael Taktikos, May 26 2006. This is very fast. *)

Range[0, 20]! CoefficientList[ Series[ Exp[ -x]/(1 - x), {x, 0, 20}], x]

dr[{n_, a1_, a2_}]:={n+1, a2, n(a1+a2)}; Transpose[NestList[dr, {0, 0, 1}, 30]][[3]] (* Harvey P. Dale, Feb 23 2013 *)

a[n_] := If[ n < 1, Boole[n == 0], Round[ n! / E]]; (* Michael Somos, Jun 01 2013 *)

a[n_] := (-1)^n HypergeometricPFQ[{- n, 1}, {}, 1]; (* Michael Somos, Jun 01 2013 *)

a[n_] := n! SeriesCoefficient[Exp[-x] /(1 - x), {x, 0, n}]; (* Michael Somos, Jun 01 2013 *)

Table[Subfactorial[n], {n, 0, 21}] (* Jean-François Alcover, Jan 10 2014 *)

RecurrenceTable[{a[n] == n*a[n - 1] + (-1)^n, a[0] == 1}, a, {n, 0, 23}] (* Ray Chandler, Jul 30 2015 *)

Subfactorial[Range[0, 20]] (* Eric W. Weisstein, Dec 31 2017 *)


(PARI) {a(n) = if( n<1, 1, n * a(n-1) + (-1)^n)}; /* Michael Somos, Mar 24 2003 */

(PARI) {a(n) = n! * polcoeff( exp(-x + x * O(x^n)) / (1 - x), n)}; /* Michael Somos, Mar 24 2003 */

(PARI) {a(n)=polcoeff(sum(m=0, n, m^m*x^m/(1+(m+1)*x+x*O(x^n))^(m+1)), n)} /* Paul D. Hanna */

(PARI) A000166=n->n!*sum(k=0, n, (-1)^k/k!) \\ M. F. Hasler, Jan 26 2012

(PARI) a(n)=if(n, round(n!/exp(1)), 1) \\ Charles R Greathouse IV, Jun 17 2012

(Python) See Hobson link.




makelist(s[n], n, 0, 12); /* Emanuele Munarini, Mar 01 2011 */


a000166 n = a000166_list !! n

a000166_list = 1 : 0 : zipWith (*) [1..]

                       (zipWith (+) a000166_list $ tail a000166_list)

-- Reinhard Zumkeller, Dec 09 2012


A000166_list, m, x = [], 1, 1

for n in range(10*2):

....x, m = x*n + m, -m

....A000166_list.append(x) # Chai Wah Wu, Nov 03 2014

(MAGMA) I:=[0, 1]; [1] cat [n le 2 select I[n] else (n-1)*(Self(n-1)+Self(n-2)): n in [1..30]]; // Vincenzo Librandi, Jan 07 2016


Cf. A000142, A002467, A003221, A000522, A000240, A000387, A000449, A000475, A129135, A092582, A000255, A002469, A159610, A068985, A068996, A047865, A038205, A008279.

For the probabilities a(n)/n!, see A053557/A053556 and A103816/A053556.

A diagonal of A008291 and A068106. A column of A008290.

A001120 has a similar recurrence.

For other derangement numbers see also A053871, A033030, A088991, A088992.

Pairwise sums of A002741 and A000757. Differences of A001277.

Cf. A101560, A101559, A000110, A101033, A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.

A diagonal in triangles A008305 and A010027.

a(n)/n! = A053557/A053556 = (N(n, n) of A103361)/(D(n, n) of A103360).

a(n) = A086764(n,0).

Row sums of A216963.

Sequence in context: A257953 A260216 A182386 * A093464 A196301 A184932

Adjacent sequences:  A000163 A000164 A000165 * A000167 A000168 A000169




N. J. A. Sloane


Minor edits by M. F. Hasler, Jan 16 2017



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Last modified January 19 14:26 EST 2019. Contains 319307 sequences. (Running on oeis4.)