

A001710


Order of alternating group A_n, or number of even permutations of n letters.
(Formerly M2933 N1179)


195



1, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000, 25545471085854720000, 562000363888803840000
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OFFSET

0,4


COMMENTS

For n >= 3, a(n1) is also the number of ways that a 3cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n).  Ahmed Fares (ahmedfares(AT)mydeja.com), Aug 14 2001
a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph.  Chad Brewbaker, Jan 31 2003
a(n1) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles.  Chad Brewbaker, Jan 31 2003
Number of increasing runs in all permutations of [n1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [3] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1).  Emeric Deutsch, Aug 28 2004
Minimum permanent over all n X n (0,1)matrices with exactly n/2 zeros.  Simone Severini, Oct 15 2004
The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0, 1, 3, 12, 60, 360, 2520, 20160, ... .  Jon Perry, Sep 20 2008
Starting (1, 3, 12, 60, ...) = binomial transform of A000153: (1, 2, 7, 32, 181, ...).  Gary W. Adamson, Dec 25 2008
First column of A092582.  Mats Granvik, Feb 08 2009
The asymptotic expansion of the higher order exponential integral E(x,m=1,n=3) ~ exp(x)/x*(1  3/x + 12/x^2  60/x^3 + 360/x^4  2520/x^5 + 20160/x^6  81440/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information.  Johannes W. Meijer, Oct 20 2009
For n>1: a(n) = A173333(n,2).  Reinhard Zumkeller, Feb 19 2010
Starting (1, 3, 12, 60, ...) = eigensequence of triangle A002260, (a triangle with k terms of (1,2,3,...) in each row given k=1,2,3,...). Example: a(6) = 360, generated from (1, 2, 3, 4, 5) dot (1, 1, 3, 12, 60) = (1 + 2 + 9 + 48 + 300).  Gary W. Adamson, Aug 02 2010
For n>=2: a(n) is the number of connected 2regular labeled graphs on (n+1) nodes (Cf. A001205).  Geoffrey Critzer, Feb 16 2011.
The Fi1 and Fi2 triangle sums of A094638 are given by the terms of this sequence (n>=1). For the definition of these triangle sums see A180662.  Johannes W. Meijer, Apr 20 2011
Also [1, 1] together with the row sums of triangle A162608.  Omar E. Pol, Mar 09 2012
a(n1) is, for n>=2, also the number of necklaces with n beads (only C_n symmetry, no turnover) with n1 distinct colors and signature c[.]^2 c[.]^(n2). This means that two beads have the same color, and for n=2 the second factor is omitted. Say, cyclic(c[1]c[1]c[2]c[3]..c[n1]), in short 1123...(n1), taken cyclically. E.g., n=2: 11, n=3: 112, n=4: 1123, 1132, 1213, n=5: 11234, 11243, 11324, 11342, 11423, 11432, 12134, 12143, 13124, 13142, 14123, 14132. See the nexttolast entry in line n>=2 of the representative necklace partition array A212359.  Wolfdieter Lang, Jun 26 2012
For m >= 3, a(m1) is the number of distinct Hamiltonian circuits in a complete simple graph with m vertices. See also A001286.  Stanislav Sykora, May 10 2014
In factorial base (A007623) these numbers have a simple pattern: 1, 1, 1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000, 5500000000, 60000000000, 660000000000, 7000000000000, 77000000000000, 800000000000000, 8800000000000000, 90000000000000000, 990000000000000000, etc. See also the formula based on this observation, given below.  Antti Karttunen, Dec 19 2015
Also (by definition) the independence number of the ntransposition graph.  Eric W. Weisstein, May 21 2017
Number of permutations of n letters containing an even number of even cycles.  Michael Somos, Jul 11 2018
Equivalent to Brewbaker's and Sykora's comments, a(n  1) is the number of undirected cycles covering n labeled vertices, hence the logarithmic transform of A002135.  Gus Wiseman, Oct 20 2018
For n >= 2 and a set of n distinct leaf labels, a(n) is the number of binary, rooted, leaflabeled tree topologies that have a caterpillar shape (column k=1 of A306364).  Noah A Rosenberg, Feb 11 2019
Also the clique covering number of the nBruhat graph.  Eric W. Weisstein, Apr 19 2019
a(n) is the number of lattices of the form [s,w] in the weak order on S_n, for a fixed simple reflection s.  Bridget Tenner, Jan 16 2020
For n > 3, a(n) = p_1^e_1*...*p_m^e_m, where p_1 = 2 and e_m = 1. There exists p_1^x where x <= e_1 such that p_1^x*p_m^e_m is a primitive Zumkeller number (A180332) and p_1^e_1*p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 3, a(n) = p_1^e_1*p_m^e_m*r, where r is relatively prime to p_1*p_m, is also a Zumkeller number.  Ivan N. Ianakiev, Mar 11 2020
For n>1, a(n) is the number of permutations of [n] that have 1 and 2 as cyclemates, that is, 1 and 2 are contained in the same cycle of a cyclic representation of permutations of [n]. For example, a(4) counts the 12 permutations with 1 and 2 as cyclemates, namely, (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2), (1 2 3) (4), (1 3 2) (4), (1 2 4 )(3), (1 4 2)(3), (1 2)(3 4), and (1 2)(3)(4). Since a(n+2)=row sums of A162608, our result readily follows.  Dennis P. Walsh, May 28 2020


REFERENCES

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 878, 20. (a), c_n^e(t=1).
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).


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Paul Barry, On the Gapsum and Gapproduct Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Camille Combe and Samuele Giraudo, Cliff operads: a hierarchy of operads on words, arXiv:2106.14552 [math.CO], 2021.
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515541, Remark 7.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 262
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Shirali Kadyrov and Farukh Mashurov, Generalized continued fraction expansions for Pi and E, arXiv:1912.03214 [math.NT], 2019.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Xah Lee, Combinatorics: Loop in n points
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 177.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 4449 [Annotated scanned copy]
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some WellKnown Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
SZ Song, SG Hwang, SH Rim, and GS Cheon, Extremes of permanents of (0,1)matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197210.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207214.
B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Alternating Group
Eric Weisstein's World of Mathematics, Bruhat Graph
Eric Weisstein's World of Mathematics, Circular Permutation
Eric Weisstein's World of Mathematics, Clique Covering Number
Eric Weisstein's World of Mathematics, Even Permutation
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Odd Permutation
Eric Weisstein's World of Mathematics, Transposition Graph
Index to divisibility sequences
Index entries for sequences related to factorial base representation
Index entries for sequences related to factorial numbers
Index entries for sequences related to groups


FORMULA

a(n) = numerator(n!/2) and A141044(n) = denominator(n!/2).
Dfinite with recurrence: a(0) = a(1) = a(2) = 1; a(n) = n*a(n1) for n>2.  Chad Brewbaker, Jan 31 2003 [Corrected by N. J. A. Sloane, Jul 25 2008]
a(0) = 0, a(1) = 1; a(n) = Sum_{k=1..n1} k*a(k).  Amarnath Murthy, Oct 29 2002
Stirling transform of a(n+1) = [1, 3, 12, 160, ...] is A083410(n) = [1, 4, 22, 154, ...].  Michael Somos, Mar 04 2004
First Eulerian transform of A000027. See A000142 for definition of FET.  Ross La Haye, Feb 14 2005
From Paul Barry, Apr 18 2005: (Start)
a(n) = 0^n + Sum_{k=0..n} (1)^(nk1)*T(n1, k)*cos(Pi*(nk1)/2)^2.
T(n,k) = abs(A008276(n, k)). (End)
E.g.f.: (2  x^2)/(2  2*x).
E.g.f. of a(n+2), n>=0, is 1/(1x)^3.
E.g.f.: 1 + sinh(log(1/(1x))).  Geoffrey Critzer, Dec 12 2010
a(n+1) = (1)^n * A136656(n,1), n>=1.
a(n) = n!/2 for n>=2 (proof from the e.g.f).  Wolfdieter Lang, Apr 30 2010
a(n) = (n2)! * t(n1), n>1, where t(n) is the nth triangular number (A000217).  Gary Detlefs, May 21 2010
a(n) = ( A000254(n)  2* A001711(n3) )/3, n>2.  Gary Detlefs, May 24 2010
O.g.f.: 1 + x*Sum_{n>=0} n^n*x^n/(1 + n*x)^n.  Paul D. Hanna, Sep 13 2011
a(n) = if n < 2 then 1, otherwise Pochhammer(n,n)/binomial(2*n,n).  Peter Luschny, Nov 07 2011
a(n) = Sum_{k=0..floor(n/2)} s(n,n2*k) where s(n,k) are Stirling number of the first kind, A048994.  Mircea Merca, Apr 07 2012
a(n1), n>=3, is M_1([2,1^(n2)])/n = (n1)!/2, with the M_1 multinomial numbers for the given n1 part partition of n. See the second to last entry in line n>=3 of A036038, and the above necklace comment by W. Lang.  Wolfdieter Lang, Jun 26 2012
G.f.: A(x) = 1 + x + x^2/(G(0)2*x) where G(k) = 1  (k+1)*x/(1  x*(k+3)/G(k+1)); (continued fraction).  Sergei N. Gladkovskii, Dec 26 2012.
G.f.: 1 + x + (Q(0)1)*x^2/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+2)*sqrt(x)/(1  sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction).  Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + (x*Q(x)x^2)/(2*(sqrt(x)+x)), where Q(x) = Sum_{n>=0} (n+1)!*x^n*sqrt(x)*(sqrt(x) + x*(n+2)).  Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x/2 + (Q(0)1)*x/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+1)*sqrt(x)/(1  sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction).  Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + x^2*G(0)/2, where G(k) = 1 + 1/(1  x/(x + 1/(k+3)/G(k+1))); (continued fraction).  Sergei N. Gladkovskii, Jun 01 2013
G.f.: 1+x + x^2*W(0), where W(k) = 1  x*(k+3)/( x*(k+3)  1/(1  x*(k+1)/( x*(k+1)  1/W(k+1) ))); (continued fraction).  Sergei N. Gladkovskii, Aug 26 2013
From Antti Karttunen, Dec 19 2015: (Start)
a(0)=a(1)=1; after which, for even n: a(n) = (n/2) * (n1)!, and for odd n: a(n) = (n1)/2 * ((n1)! + (n2)!). [The formula was empirically found after viewing these numbers in factorial base, A007623, and is easily proved by considering formulas from Lang (Apr 30 2010) and Detlefs (May 21 2010) shown above.]
For n >= 1, a(2*n+1) = a(2*n) + A153880(a(2*n)). [Follows from above.] (End)
Inverse Stirling transform of a(n) is (1)^(n1)*A009566(n).  Anton Zakharov, Aug 07 2016
a(n) ~ sqrt(Pi/2)*n^(n+1/2)/exp(n).  Ilya Gutkovskiy, Aug 07 2016
a(n) = A006595(n1)*n/A000124(n) for n>=2.  Anton Zakharov, Aug 23 2016
a(n) = A001563(n1)  A001286(n1) for n>=2.  Anton Zakharov, Sep 23 2016
From Peter Bala, May 24 2017: (Start)
The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (x  1)*A(x) + 1  x^2 = 0.
G.f.: A(x) = 1 + x + x^2/(1  3*x/(1  x/(1  4*x/(1  2*x/(1  5*x/(1  3*x/(1  ...  (n + 2)*x/(1  n*x/(1  ... ))))))))) (apply Stokes, 1982).
A(x) = 1 + x + x^2/(1  2*x  x/(1  3*x/(1  2*x/(1  4*x/(1  3*x/(1  5*x/(1  ...  n*x/(1  (n+2)*x/(1  ... ))))))))). (End)
H(x) = (1  (1 + x)^(2)) / 2 = x  3*x^2/2! + 12*x^3/3!  ..., an e.g.f. for the signed sequence here (n!/2!), ignoring the first two terms, is the compositional inverse of G(x) = (1  2*x)^(1/2)  1 = x + 3*x^2/2! + 15*x^3/3! + ..., an e.g.f. for A001147. Cf. A094638. H(x) is the e.g.f. for the sequence (1)^m * m!/2 for m = 2,3,4,... . Cf. A001715 for n!/3! and A001720 for n!/4!. Cf. columns of A094587, A173333, and A213936 and rows of A138533.  Tom Copeland, Dec 27 2019


EXAMPLE

G.f. = 1 + x + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 360*x^6 + 2520*x^7 + ...


MAPLE

seq(mul(k, k=3..n), n=0..20); # Zerinvary Lajos, Sep 14 2007


MATHEMATICA

a[n_]:= If[n > 2, n!/2, 1]; Array[a, 21, 0]
a[n_]:= If[n<3, 1, n*a[n1]]; Array[a, 21, 0]; (* Robert G. Wilson v, Apr 16 2011 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[(2x^2)/(22x), {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[1 +Sinh[Log[1x]], {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
Numerator[Range[0, 20]!/2] (* Eric W. Weisstein, May 21 2017 *)
Table[GroupOrder[AlternatingGroup[n]], {n, 0, 20}] (* Eric W. Weisstein, May 21 2017 *)


PROG

(Magma) [1] cat [Order(AlternatingGroup(n)): n in [1..20]]; // Arkadiusz Wesolowski, May 17 2014
(PARI) {a(n) = if( n<2, n>=0, n!/2)};
(PARI) a(n)=polcoeff(1+x*sum(m=0, n, m^m*x^m/(1+m*x+x*O(x^n))^m), n) \\ Paul D. Hanna
(PARI) A001710=n>n!\2+(n<2) \\ M. F. Hasler, Dec 01 2013
(Scheme, using memoizationmacro definec for which an implementation can be found in http://oeis.org/wiki/Memoization )
(definec (A001710 n) (cond ((<= n 2) 1) (else (* n (A001710 ( n 1))))))
;; Antti Karttunen, Dec 19 2015


CROSSREFS

Cf. A000142, A000153, A000255, A001147, A001286, A001720, A002135, A002260, A007623, A007717, A049444, A049459, A093468, A094587, A094638, A138533, A153880, A173333, A213936, A215771, A319225, A319226, A320655.
a(n+1)= A046089(n, 1), n >= 1 (first column of triangle), A161739 (q(n) sequence).
Bisections are A002674 and A085990 (essentially).
Row 3 of A265609 (essentially).
Row sums of A307429.
Sequence in context: A062569 A089057 A077134 * A105752 A177138 A053532
Adjacent sequences: A001707 A001708 A001709 * A001711 A001712 A001713


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001
Further terms from Simone Severini, Oct 15 2004


STATUS

approved



