A000186 -id:A000186 - cp's OEIS frontend

A000179 Ménage numbers: a(0) = 1, a(1) = -1, and for n >= 2, a(n) = number of permutations s of [0, ..., n-1] such that s(i) != i and s(i) != i+1 (mod n) for all i.

1, -1, 0, 1, 2, 13, 80, 579, 4738, 43387, 439792, 4890741, 59216642, 775596313, 10927434464, 164806435783, 2649391469058, 45226435601207, 817056406224416, 15574618910994665, 312400218671253762, 6577618644576902053, 145051250421230224304, 3343382818203784146955, 80399425364623070680706, 2013619745874493923699123

Offset: 0

N. J. A. Sloane

According to rook theory, John Riordan considered a(1) to be -1. - Vladimir Shevelev, Apr 02 2010
This is also the value that the formulas of Touchard and of Wyman and Moser give and is compatible with many recurrences. - William P. Orrick, Aug 31 2020
Or, for n >= 3, the number of 3 X n Latin rectangles the second row of which is full cycle with a fixed order of its elements, e.g., the cycle (x_2,x_3,...,x_n,x_1) with x_1 < x_2 < ... < x_n. - Vladimir Shevelev, Mar 22 2010
Muir (p. 112) gives essentially this recurrence (although without specifying any initial conditions). Compare A186638. - N. J. A. Sloane, Feb 24 2011
Sequence discovered by Touchard in 1934. - L. Edson Jeffery, Nov 13 2013
Although these are also known as Touchard numbers, the problem was formulated by Lucas in 1891, who gave the recurrence formula shown below. See Cerasoli et al., 1988. - Stanislav Sykora, Mar 14 2014
An equivalent problem was formulated by Tait; solutions to Tait's problem were given by Muir (1878) and Cayley (1878). - William P. Orrick, Aug 31 2020
From Vladimir Shevelev, Jun 25 2015: (Start)
According to the ménage problem, 2*n!*a(n) is the number of ways of seating n married couples at 2*n chairs around a circular table, men and women in alternate positions, so that no husband is next to his wife.
It is known [Riordan, ch. 7] that a(n) is the number of arrangements of n non-attacking rooks on the positions of the 1's in an n X n (0,1)-matrix A_n with 0's in positions (i,i), i = 1,...,n, (i,i+1), i = 1,...,n-1, and (n,1). This statement could be written as a(n) = per(A_n). For example, A_5 has the form
    001*11
    1*0011
    11001*                        (1)
    11*100
    0111*0,
where 5 non-attacking rooks are denoted by {1*}.
We can indicate a one-to-one correspondence between arrangements of n non-attacking rooks on the 1's of a matrix A_n and arrangements of n married couples around a circular table by the rules of the ménage problem, after the ladies w_1, w_2, ..., w_n have taken the chairs numbered
    2*n, 2, 4, ..., 2*n-2         (2)
respectively. Suppose we consider an arrangement of rooks: (1,j_1), (2,j_2), ..., (n,j_n). Then the men m_1, m_2, ..., m_n took chairs with numbers
    2*j_i - 3 (mod 2*n),          (3)
where the residues are chosen from the interval[1,2*n]. Indeed {j_i} is a permutation of 1,...,n. So {2*j_i-3}(mod 2*n) is a permutation of odd positive integers <= 2*n-1. Besides, the distance between m_i and w_i cannot be 1. Indeed, the equality |2*(j_i-i)-1| = 1 (mod 2*n) is possible if and only if either j_i=i or j_i=i+1 (mod n) that correspond to positions of 0's in matrix A_n.
For example, in the case of positions of {1*} in(1) we have j_1=3, j_2=1, j_3=5, j_4=2, j_5=4. So, by(2) and (3) the chairs 1,2,...,10 are taken by m_4, w_2, m_1, w_3, m_5, w_4, m_3, w_5, m_2, w_1, respectively. (End)
The first 20 terms of this sequence were calculated in 1891 by E. Lucas (see [Lucas, p. 495]). - Peter J. C. Moses, Jun 26 2015
From Ira M. Gessel, Nov 27 2018: (Start)
If we invert the formula
  Sum_{ n>=0 } u_n z^n = ((1-z)/(1+z)) F(z/(1+z)^2)
that Don Knuth mentions (see link) (i.e., set x=z/(1+z)^2 and solve for z in terms of x), we get a formula for F(z) = Sum_{n >= 0} n! z^n as a sum with all positive coefficients of (almost) powers of the Catalan number generating function.
The exact formula is (5) of the Yiting Li article.
This article also gives a combinatorial proof of this formula (though it is not as simple as one might want). (End)

			a(2) = 0; nothing works. a(3) = 1; (201) works. a(4) = 2; (2301), (3012) work. a(5) = 13; (20413), (23401), (24013), (24103), (30412), (30421), (34012), (34021), (34102), (40123), (43012), (43021), (43102) work.

W. W. R. Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th Ed. Dover, p. 50.
M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Nicola Zanichelli Editore, Bologna 1988, Chapter 3, p. 78.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 185, mu(n).
Kaplansky, Irving and Riordan, John, The probleme des menages, Scripta Math. 12, (1946). 113-124. See u_n.
E. Lucas, Théorie des nombres, Paris, 1891, pp. 491-495.
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 1, p 256.
T. Muir, A Treatise on the Theory of Determinants. Dover, NY, 1960, Sect. 132, p. 112. - N. J. A. Sloane, Feb 24 2011
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 197.
V. S. Shevelev, Reduced Latin rectangles and square matrices with equal row and column sums, Diskr. Mat. (J. of the Akademy of Sciences of Russia) 4(1992), 91-110. - Vladimir Shevelev, Mar 22 2010
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. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff.
J. Touchard, Permutations discordant with two given permutations, Scripta Math., 19 (1953), 108-119.
J. H. van Lint, Combinatorial Theory Seminar, Eindhoven University of Technology, Springer Lecture Notes in Mathematics, Vol. 382, 1974. See page 10.

Seiichi Manyama, Table of n, a(n) for n = 0..450 (terms 0..100 from T. D. Noe)
M. A. Alekseyev, Weighted de Bruijn Graphs for the Menage Problem and Its Generalizations. Lecture Notes in Computer Science 9843 (2016), 151-162. doi:10.1007/978-3-319-44543-4_12 arXiv:1510.07926, [math.CO], 2015-2016.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135.
Kenneth P. Bogart and Peter G. Doyle, Nonsexist solution of the ménage problem, Amer. Math. Monthly 93 (1986), no. 7, 514-519.
A. Cayley, On a problem of arrangements, Proceedings of the Royal Society of Edinburgh 9 (1878) 338-342.
A. Cayley, On a problem of arrangements, Proceedings of the Royal Society of Edinburgh 9 (1878) 338-342.
A. Cayley, On Mr Muir's discussion of Professor Tait's problem of arrangements, Proceedings of the Royal Society of Edinburgh 9 (1878) 388-391.
A. Cayley, On Mr Muir's discussion of Professor Tait's problem of arrangements, Proceedings of the Royal Society of Edinburgh 9 (1878) 388-391.
J. Dutka, On the Probleme des Menages, Mathem. Conversat. (2001) 277-287, reprinted from Math. Intell. 8 (1986) 18-25
A. de Gennaro, How many latin rectangles are there?, arXiv:0711.0527 [math.CO] (2007), see p. 2.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 372
Nick Hobson, Python program for this sequence.
Peter Kagey, Ranking and Unranking Restricted Permutations, arXiv:2210.17021 [math.CO], 2022.
Irving Kaplansky, Solution of the "Problème des ménages", Bull. Amer. Math. Soc. 49, (1943). 784-785.
Irving Kaplansky, Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc., 50 (1944), 906-914.
I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124. [Scan of annotated copy]
S. M. Kerawala, Asymptotic solution of the "Probleme des menages, Bull. Calcutta Math. Soc., 39 (1947), 82-84. [Annotated scanned copy]
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 221.
D. E. Knuth, Comments on A000179, Nov 25 2018 - Nov 27 2018.
D. E. Knuth, Donald Knuth's 24th Annual Christmas Lecture: Dancing Links, Stanfordonline, Video published on YouTube, Dec 12, 2018.
A. R. Kräuter, Über die Permanente gewisser zirkulärer Matrizen..., Séminaire Lotharingien de Combinatoire, B11b (1984), 11 pp.
Yiting Li, Ménage Numbers and Ménage Permutations, J. Int. Seq. 18 (2015) 15.6.8
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 495.
T. Muir, On Professor Tait's problem of arrangement, Proceedings of the Royal Society of Edinburgh 9 (1878) 382-387.
T. Muir, On Professor Tait's problem of arrangement, Proceedings of the Royal Society of Edinburgh 9 (1878) 382-387.
Vladimir Shevelev and Peter J. C. Moses, The ménage problem with a known mathematician, arXiv:1101.5321 [math.CO], 2011, 2015.
Vladimir Shevelev and Peter J. C. Moses, Alice and Bob go to dinner: A variation on menage, INTEGERS, Vol. 16 (2016), #A72.
R. J. Stones, S. Lin, X. Liu and G. Wang, On Computing the Number of Latin Rectangles, Graphs and Combinatorics (2016) 32:1187-1202; DOI 10.1007/s00373-015-1643-1.
H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff. [Annotated scanned copy]
J. Touchard, Théorie des substitutions. Sur un problème de permutations, C. R. Acad. Sci. Paris 198 (1934), 631-633.
Eric Weisstein's World of Mathematics, Married Couples Problem.
Eric Weisstein's World of Mathematics, Rooks Problem.
Wikipedia, Menage problem.
M. Wyman and L. Moser, On the problème des ménages, Canad. J. Math., 10 (1958), 468-480.
D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.

Diagonal of A058087. Also a diagonal of A008305.
Cf. A000904, A059375, A102761, A000186, A094047, A067998, A033999, A258664, A258665, A258666, A258667, A258673, A259212, A213234, A000023.
A000179, A102761, and A335700 are all essentially the same sequence but with different conventions for the initial terms a(0) and a(1). - N. J. A. Sloane, Aug 06 2020

import Data.List (zipWith5)
a000179 n = a000179_list !! n
a000179_list = 1 : -1 : 0 : 1 : zipWith5
   (\v w x y z -> (x * y + (v + 2) * z - w) `div` v) [2..] (cycle [4,-4])
   (drop 4 a067998_list) (drop 3 a000179_list) (drop 2 a000179_list)
-- Reinhard Zumkeller, Aug 26 2013

A000179:= n ->add ((-1)^k*(2*n)*binomial(2*n-k,k)*(n-k)!/(2*n-k), k=0..n); # for n >= 1
U:= proc(n) local k; add( (2*n/(2*n-k))*binomial(2*n-k,k)*(n-k)!*(x-1)^k, k=0..n); end; W := proc(r,s) coeff( U(r),x,s ); end; A000179 := n->W(n,0); # valid for n >= 1

a[n_] := 2*n*Sum[(-1)^k*Binomial[2*n - k, k]*(n - k)!/(2*n - k), {k, 0, n}]; a[0] = 1; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 05 2012, from 2nd formula *)

\\ 3 programs adapted to a(1) = -1 by Hugo Pfoertner, Aug 31 2020

{a(n) = my(A); if( n, A = vector(n,i,i-2); for(k=4, n, A[k] = (k * (k - 2) * A[k-1] + k * A[k-2] - 4 * (-1)^k) / (k-2)); A[n], 1)};/* Michael Somos, Jan 22 2008 */

a(n)=if(n>1, round(2*n*exp(-2)*besselk(n, 2)), 1-2*n) \\ Charles R Greathouse IV, Nov 03 2014

{a(n) = my(A); if( n, A = vector(n,i,i-2); for(k=5, n, A[k] = k * A[k-1] + 2 * A[k-2] + (4-k) * A[k-3] - A[k-4]); A[n], 1)} /* Michael Somos, May 02 2018 */

from math import comb, factorial
def A000179(n): return 1 if n == 0 else sum((-2*n if k & 1 else 2*n)*comb(m:=2*n-k,k)*factorial(n-k)//m for k in range(n+1)) # Chai Wah Wu, May 27 2022

a(n) = ((n^2-2*n)*a(n-1) + n*a(n-2) - 4*(-1)^n)/(n-2) for n >= 3.
a(n) = A059375(n)/(2*n!) for n >= 2.
a(n) = Sum_{k=0..n} (-1)^k*(2*n)*binomial(2*n-k, k)*(n-k)!/(2*n-k) for n >= 1. - Touchard (1934)
G.f.: ((1-x)/(1+x))*Sum_{n>=0} n!*(x/(1+x)^2)^n. - Vladeta Jovovic, Jun 26 2007
a(2^k+2) == 0 (mod 2^k); for k >= 2, a(2^k) == 2(mod 2^k). - Vladimir Shevelev, Jan 14 2011
a(n) = round( 2*n*exp(-2)*BesselK(n,2) ) for n > 1. - Mark van Hoeij, Oct 25 2011
a(n) ~ (n/e)^n * sqrt(2*Pi*n)/e^2. - Charles R Greathouse IV, Jan 21 2016
0 = a(n)*(-a(n+2) +a(n+4)) +a(n+1)*(+a(n+1) +a(n+2) -3*a(n+3) -5*a(n+4) +a(n+5)) +a(n+2)*(+2*a(n+2) +3*a(n+3) -3*a(n+4)) +a(n+3)*(+2*a(n+3) +a(n+4) -a(n+5)) +a(n+4)*(+a(n+4)), for all n>1. If a(-2..1) = (0, -1, 2, -1) then also true for those values of n. - Michael Somos, Apr 29 2018
D-finite with recurrence: 0 = a(n) +n*a(n+1) -2*a(n+2) +(-n-4)*a(n+3) +a(n+4), for all n in Z where a(n) = a(-n) for all n in Z and a(0) = 2, a(1) = -1. - Michael Somos, May 02 2018
a(n) = Sum_{k=0..n} A213234(n,k) * A000023(n-2*k) = Sum_{k=0..n} (-1)^k * n/(n-k) * binomial(n-k, k) * (n-2*k)! Sum_{j=0..n-2*k} (-2)^j/j! for n >= 1. [Wyman and Moser (1958)]. - William P. Orrick, Jun 25 2020
a(k+4*p) - 2*a(k+2*p) + a(k) is divisible by p, for any k > 0 and any prime p. - Mark van Hoeij, Jan 11 2022

More terms from James Sellers, May 02 2000
Additional comments from David W. Wilson, Feb 18 2003
a(1) changed to -1 at the suggestion of Don Knuth. - N. J. A. Sloane, Nov 26 2018

A094047 Number of seating arrangements of n couples around a round table (up to rotations) so that each person sits between two people of the opposite sex and no couple is seated together.

Original entry on oeis.org

0, 0, 2, 12, 312, 9600, 416880, 23879520, 1749363840, 159591720960, 17747520940800, 2363738855385600, 371511874881100800, 68045361697964851200, 14367543450324474009600, 3464541314885011705344000, 946263209467217020194816000, 290616691739323132839591936000

Offset: 1

Matthijs Coster, Apr 29 2004

nonn

Also, the number of Hamiltonian directed circuits in the crown graph of order n.

Or the number of those 3 X n Latin rectangles (cf. A000186) the second row of which is a full cycle. - Vladimir Shevelev, Mar 22 2010

V. S. Shevelev, Reduced Latin rectangles and square matrices with equal row and column sums, Diskr.Mat.(J. of the Akademy of Sciences of Russia) 4(1992),91-110.

Seiichi Manyama, Table of n, a(n) for n = 1..253
M. A. Alekseyev, Weighted de Bruijn Graphs for the Menage Problem and Its Generalizations. Lecture Notes in Computer Science 9843 (2016), 151-162. doi:10.1007/978-3-319-44543-4_12, arXiv:1510.07926, [math.CO], 2015-2016.
H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Crown Graph
Eric Weisstein's World of Mathematics, Hamiltonian Cycle

Cf. A059375 (rotations are counted as different).

Cf. A000179, A000186, A114939, A137729, A306496.

A094047 := proc(n)
    if n < 3 then
        0;
    else
        (-1)^n*2*(n-1)!+n!*add( (-1)^j*(n-j-1)!*binomial(2*n-j-1,j),j=0..n-1) ;
    end if;
end proc: # R. J. Mathar, Nov 02 2015

Join[{0},Table[(-1)^n 2(n-1)!+n!Sum[(-1)^j (n-j-1)!Binomial[2n-j-1,j],{j,0,n-1}],{n,2,20}]] (* Harvey P. Dale, Mar 07 2012 *)

For n>1, a(n) = (-1)^n * 2 * (n-1)! + n! * Sum_{j=0..n-1} (-1)^j * (n-j-1)! * binomial(2*n-j-1,j). - Max Alekseyev, Feb 10 2008
a(n) = A059375(n) / (2*n) = A000179(n) * (n-1)!.
Conjecture: a(n) +(-n^2+2*n-3)*a(n-1) -(n-2)*(n^2-3*n+5)*a(n-2) -3*(n-2)*(n-3)*a(n-3) +(n-2)*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Nov 02 2015
Conjecture: (-n+2)*a(n) +(n-1)*(n^2-3*n+3)*a(n-1) +(n-2)*(n-1)*(n^2-3*n+3)*a(n-2) +(n-2)*(n-3)*(n-1)^2*a(n-3)=0. - R. J. Mathar, Nov 02 2015
a(n) = (n-1) * (n * (a(n-1) + a(n-2)) - 4 * (-1)^n * (n-3)!) for n > 3. - Seiichi Manyama, Jan 18 2020
a(n) = 2 * A306496(n). - Alois P. Heinz, Jun 19 2022

Better definition from Joel B. Lewis, Jun 30 2007

Formula and further terms from Max Alekseyev, Feb 10 2008

A000512 Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 3, where equivalence is defined by row and column permutations.

Original entry on oeis.org

0, 0, 1, 1, 2, 7, 16, 51, 224, 1165, 7454, 56349, 481309, 4548786, 46829325, 519812910, 6177695783, 78190425826, 1049510787100, 14886252250208, 222442888670708, 3492326723315796, 57468395960854710, 989052970923320185, 17767732298980160822, 332572885090541084172, 6475438355244504235759, 130954580036269713385884

Offset: 1

Eric Rogoyski

Also, isomorphism classes of bicolored cubic bipartite graphs, where isomorphism cannot exchange the colors.

			n=4: every matrix with 3 1's in each row and column can be transformed by permutation of rows (or columns) into {1110,1101,1011,0111}, therefore a(4)=1. - _Michael Steyer_, Feb 20 2003

A. Burgess, P. Danziger, E. Mendelsohn, B. Stevens, Orthogonally Resolvable Cycle Decompositions, 2013; http://www.math.ryerson.ca/~andrea.burgess/OCD-submit.pdf
Goulden and Jackson, Combin. Enum., Wiley, 1983 p. 284.

Index entries for sequences related to Latin squares and rectangles

Column k=3 of A133687.
Cf. A000186, A000513, A000840, A001501, A008325, A232215.
A079815 may be an erroneous version of this, or it may have a slightly different (as yet unknown) definition. - N. J. A. Sloane, Sep 04 2010.

Definition corrected by Brendan McKay, May 28 2006
a(1)-a(12) checked by Brendan McKay, Aug 27 2010
Terms a(15) and beyond from Andrew Howroyd, Apr 01 2020

A102761 Same as A000179, except that a(0) = 2.

Original entry on oeis.org

2, -1, 0, 1, 2, 13, 80, 579, 4738, 43387, 439792, 4890741, 59216642, 775596313, 10927434464, 164806435783, 2649391469058, 45226435601207, 817056406224416, 15574618910994665, 312400218671253762, 6577618644576902053, 145051250421230224304, 3343382818203784146955, 80399425364623070680706, 2013619745874493923699123

Offset: 0

N. J. A. Sloane, Apr 04 2010, following a suggestion from Vladimir Shevelev

For any integer n>=0, 2 * Integral_{t=-2..2} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-1..1} T_n(z)*exp(-2*z)*dz = a(n)*exp(2) - A300484(n)*exp(-2). - Max Alekseyev, Mar 08 2018

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 197.

Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Λ_n^3 and Λ_n(α,β,γ), arXiv:1104.4051 [math.CO], 2011.

Row m=2 in A300481.
Cf. A000023, A000179, A000186, A300484.
A000179, A102761, and A335700 are all essentially the same sequence but with different conventions for the initial terms a(0) and a(1). - N. J. A. Sloane, Aug 06 2020

{ A102761(n) = subst( serlaplace( 2*polchebyshev(n, 1, (x-2)/2)), x, 1); } \\ Max Alekseyev, Mar 06 2018

a(n) = Sum_{i=0..n} A127672(n,i) * A000023(i). - Max Alekseyev, Mar 06 2018
a(n) = A300481(2,n) = A300480(-2,n). - Max Alekseyev, Mar 06 2018
a(n) = A335391(0,n) (Touchard). - William P. Orrick, Aug 29 2020

Changed a(0)=2 (making the sequence more consistent with existing formulae) by Max Alekseyev, Mar 06 2018

A000321 H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.

Original entry on oeis.org

1, -1, -1, 5, 1, -41, 31, 461, -895, -6481, 22591, 107029, -604031, -1964665, 17669471, 37341149, -567425279, -627491489, 19919950975, 2669742629, -759627879679, 652838174519, 31251532771999, -59976412450835, -1377594095061119, 4256461892701199, 64623242860354751

Offset: 0

N. J. A. Sloane

Binomial transform gives A067994. Inverse binomial transform gives A062267(n)*(-1)^n. - Vladimir Reshetnikov, Oct 11 2016

The congruence a(n+k) == (-1)^k*a(n) (mod k) holds for all n and k. It follows that for even k the sequence obtained by reducing a(n) modulo k is purely periodic with period a divisor of k, while for odd k the sequence obtained by reducing a(n) modulo k is purely periodic with period a divisor of 2*k. See A047974. - Peter Bala, Apr 10 2023

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 209.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..732 (terms 0..200 from T. D. Noe)
Koichi, Yamamoto, An asymptotic series for the number of three-line Latin rectangles, J. Math. Soc. Japan 1, (1950). 226-241.
Index entries for sequences related to Hermite polynomials

Cf. A000186, A062267, A144141, A293604.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 09 2018

Table[HermiteH[n, -1/2], {n, 0, 25}] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Table[(-2)^n HypergeometricU[-n/2, 1/2, 1/4], {n, 0, 25}] (* Benedict W. J. Irwin, Oct 17 2017 *)

N=66;  x='x+O('x^N);
egf=exp(-x-x^2);  Vec(serlaplace(egf))
/* Joerg Arndt, Mar 07 2013 */

vector(50, n, n--; sum(k=0, n/2, (-1)^(n-k)*k!*binomial(n, k)*binomial(n-k, k))) \\ Altug Alkan, Oct 22 2015

a(n) = polhermite(n, -1/2); \\ Michel Marcus, Oct 12 2016

from sympy import hermite
def a(n): return hermite(n, -1/2) # Indranil Ghosh, May 26 2017

E.g.f.: exp(-x-x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*k!*C(n, k)*C(n-k, k).
a(n) = - a(n-1) - 2*(n-1)*a(n-2), a(0) = 1, a(1) = -1.
A000186(n) ~ n!^2*exp(1)^(-3)*(a(0) + a(1)/n + a(2)/(2*[n]2) + ... + a(k)/(k!*[n]_k) + ...), where [n]_k = n*(n-1)*...*(n-k + 1), [n]_0 = 1. - _Vladeta Jovovic, Apr 30 2001
a(n) = Sum_{k=0..n} (-1)^(2*n-k)*C(k,n-k)*n!/k!. - Paul Barry, Oct 08 2007, corrected by Altug Alkan, Oct 22 2015
E.g.f.: 1 - x*(1 - E(0) )/(1+x) where E(k) = 1 - (1+x)/(k+1)/(1-x/(x+1/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2013
E.g.f.: -x/Q(0) where Q(k) = 1 - (1+x)/(1 - x/(x - (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013
G.f.: 1/(x*Q(0)), where Q(k) = 1 + 1/x + 2*(k+1)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 21 2013
a(n) = (-2)^n * U(-n/2, 1/2, 1/4), where U is the confluent hypergeometric function. - Benedict W. J. Irwin, Oct 17 2017
E.g.f.: Product_{k>=1} (1 + (-x)^k)^(mu(k)/k). - Ilya Gutkovskiy, May 26 2019

Formulae and more terms from Vladeta Jovovic, Apr 30 2001

cp's OEIS Frontend

A046744 Erroneous version of A000186.

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A000179 Ménage numbers: a(0) = 1, a(1) = -1, and for n >= 2, a(n) = number of permutations s of [0, ..., n-1] such that s(i) != i and s(i) != i+1 (mod n) for all i.

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A094047 Number of seating arrangements of n couples around a round table (up to rotations) so that each person sits between two people of the opposite sex and no couple is seated together.

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A000512 Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 3, where equivalence is defined by row and column permutations.

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A102761 Same as A000179, except that a(0) = 2.

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A000321 H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.

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A174564 Let J_n be n X n matrix which contains 1's only, I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (2,3,...,n,1). Then a(n) is the number of (0,1) n X n matrices A<=J_n-I-P with exactly two 1's in every row and column.

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