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A001710 Order of alternating group A_n, or number of even permutations of n letters.
(Formerly M2933 N1179)
136
1, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

For n >= 3, a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.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(n-1) 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 [n-1] (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 2-regular 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(n-1) is, for n>=2, also the number of necklaces with n beads (only C_n symmetry, no turnover) with n-1 distinct colors and signature c[.]^2 c[.]^(n-2). 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[n-1]), in short 1123...(n-1), 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 next to last entry in line n>=2 of the representative necklace partition array A212359. - Wolfdieter Lang, Jun 26 2012

For m>=3, a(m-1) 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

REFERENCES

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 87-8, 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

Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 262

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Xah Lee, Combinatorics: Loop in n points

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7

S-Z Song, S-G Hwang, S-H Rim, G-S Cheon, Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197-210.

Eric Weisstein's World of Mathematics, Alternating Group

Eric Weisstein's World of Mathematics, Circular Permutation

Eric Weisstein's World of Mathematics, Even Permutation

Eric Weisstein's World of Mathematics, Hamiltonian Cycle

Eric Weisstein's World of Mathematics, Odd Permutation

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).

a(0) = a(1) = a(2) = 1; a(n) = n*a(n-1) 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..n-1} 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

a(n) = sum{k=0..n, (-1)^(n-k-1)*T(n-1, k)*cos(Pi(n-k-1)/2)^2}+0^n; T(n,k)=abs(A008276(n, k)). - Paul Barry, Apr 18 2005

E.g.f.: (2-x^2)/(2-2*x). E.g.f. of a(n+2), n>=0, is 1/(1-x)^3.

E.g.f.: 1+sinh(log(1/(1-x))). - Geoffrey Critzer, Dec 12 2010

a(n+1) = A136656(n,1)*(-1)^n, n>=1.

a(n) = n!/2 for n>=2 (proof from the e.g.f). - Wolfdieter Lang, Apr 30 2010

a(n) = (n-2)! * t(n-1), n>1, where t(n) = A000217(n)..the n-th triangular number. - Gary Detlefs, May 21 2010

a(n) = ( A000254(n) - 2* A001711(n-3) )/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 else pochhammer(n,n)/binomial(2n,n). - Peter Luschny, Nov 07 2011

a(n) = Sum_{k=0..floor(n/2)} s(n,n-2*k) where s(n,k) are Stirling number of the first kind, A048994. - Mircea Merca, Apr 07 2012

a(n-1), n>=3, is M_1([2,1^(n-2)])/n = (n-1)!/2, with the M_1 multinomial numbers for the given n-1 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) * (n-1)!, and for odd n: a(n) = (n-1)/2 * ((n-1)! + (n-2)!). [The formula was empirically found after viewing these numbers in factorial base, A007623, and is easily proven by considering formulas from Lang (Apr 30 2010) and Detlefs (May 21 2010) shown above.]

For n >= 1, a(2n+1) = a(2n) + A153880(a(2n)). [Follows from above.] (End)

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[n - 1]]; Array[a, 21, 0]; (* Robert G. Wilson v, Apr 16 2011 *)

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (2 - x^2) / (2 - 2 x), {x, 0, n}]]; (* Michael Somos, May 22 2014 *)

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 + Sinh[ -Log[1 - x]], {x, 0, n}]]; (* Michael Somos, May 22 2014 *)

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 memoization-macro 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, A001286, A049444, A049459. a(n+1)= A046089(n, 1), n >= 1 (first column of triangle), A000153, A161739 (q(n) sequence), A093468, A002260.

Row 3 of A265609 (essentially).

Bisections are A002674 and A085990 (essentially).

Cf. also A007623, A153880.

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, Apr 30 1991

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001

Further from Simone Severini, Oct 15 2004

STATUS

approved

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Last modified July 30 20:16 EDT 2016. Contains 275185 sequences.