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A001710 Order of alternating group A_n, or number of even permutations of n letters.

Original entry on oeis.org

1, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000, 25545471085854720000, 562000363888803840000
Offset: 0

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Author

N. J. A. Sloane

Keywords

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
Also (by definition) the independence number of the n-transposition 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, leaf-labeled 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 n-Bruhat 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 cycle-mates, 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 cycle-mates, 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

Examples

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

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

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.

Programs

  • Magma
    [1] cat [Order(AlternatingGroup(n)): n in [1..20]]; // Arkadiusz Wesolowski, May 17 2014
  • 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)/(2-2x), {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 *)
Numerator[Range[0, 20]!/2] (* Eric W. Weisstein, May 21 2017 *)
Table[GroupOrder[AlternatingGroup[n]], {n, 0, 20}] (* Eric W. Weisstein, May 21 2017 *)
  • 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
  • Python
    from math import factorial
def A001710(n): return factorial(n)>>1 if n > 1 else 1 # Chai Wah Wu, Feb 14 2023
  • SageMath
    def A001710(n): return (factorial(n) +int(n<2))//2
[A001710(n) for n in range(31)] # G. C. Greubel, Sep 28 2024
  • 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

Formula

a(n) = numerator(n!/2) and A141044(n) = denominator(n!/2).
D-finite with recurrence: 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
From Paul Barry, Apr 18 2005: (Start)
a(n) = 0^n + Sum_{k=0..n} (-1)^(n-k-1)*T(n-1, k)*cos(Pi*(n-k-1)/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/(1-x)^3.
E.g.f.: 1 + sinh(log(1/(1-x))). - 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) = (n-2)! * t(n-1), n>1, where t(n) is the n-th triangular number (A000217). - 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, otherwise Pochhammer(n,n)/binomial(2*n,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 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)^(n-1)*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(n-1)*n/A000124(n) for n>=2. - Anton Zakharov, Aug 23 2016
a(n) = A001563(n-1) - A001286(n-1) 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
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 2*(e-1).
Sum_{n>=0} (-1)^n/a(n) = 2/e. (End)

Extensions

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

A079484 a(n) = (2n-1)!! * (2n+1)!!, where the double factorial is A006882.

Original entry on oeis.org

1, 3, 45, 1575, 99225, 9823275, 1404728325, 273922023375, 69850115960625, 22561587455281875, 9002073394657468125, 4348001449619557104375, 2500100833531245335015625, 1687568062633590601135546875, 1321365793042101440689133203125
Offset: 0

Views

Author

Benoit Cloitre, Jan 17 2003

Keywords

Comments

a(n) is the determinant of M(2n+1) where M(k) is the k X k matrix with m(i,j)=j if i+j=k m(i,j)=i otherwise. - Adapted to offset 0, Rainer Rosenthal, Jun 19 2024
In the following two comments on the calculation of the terms using permanents, offset 1 is assumed. In the corresponding PARI code, this is implemented with offset 0. - Hugo Pfoertner, Jun 23 2024
(-1)^n*a(n)/2^(2n-1) is the permanent of the (m X m) matrix {1/(x_i-y_j), 1<=i<=m, 1<=j<=m}, where x_1,x_2,...,x_m are the zeros of x^m-1 and y_1,y_2,...,y_m the zeros of y^m+1 and m=2n-1.
In 1881, R. F. Scott posed a conjecture that the absolute value of permanent of square matrix with elements a(i,j)= (x_i - y_j)^(-1), where x_1,...,x_n are roots of x^n=1, while y_1,...,y_n are roots of y^n=-1, equals a((n-1)/2)/2^n, if n>=1 is odd, and 0, if n>=2 is even. After a century (in 1979), the conjecture was proved by H. Minc. - Vladimir Shevelev, Dec 01 2013
a(n) is the number of permutations in S_{2n+1} in which all cycles have odd length. - José H. Nieto S., Jan 09 2012
Number of 3-bundled increasing bilabeled trees with 2n labels. - Markus Kuba, Nov 18 2014
a(n) is the number of rooted, binary, leaf-labeled topologies with 2n+2 leaves that have n+1 cherry nodes. - Noah A Rosenberg, Feb 12 2019

Examples

			G.f. = 1 + 3*x + 45*x^2 + 1575*x^3 + 99225*x^4 + 9823275*x^5 + ...
M(5) =
  [1, 2, 3, 1, 5]
  [1, 2, 2, 4, 5]
  [1, 3, 3, 4, 5]
  [4, 2, 3, 4, 5]
  [1, 2, 3, 4, 5].
Integral_{x=0..oo} x^3*BesselK(1, sqrt(x)) = 1575*Pi. - _Olivier Gérard_, May 20 2009

References

  • Miklós Bóna, A walk through combinatorics, World Scientific, 2006.

Links

Crossrefs

Cf. A001818, A000165.
Bisection of A000246, A053195, |A013069|, |A046126|. Cf. A000909.
Cf. A001044, A010791, |A129464|, A114779, are also values of similar moments.
Equals the row sums of A162005.
Cf. A316728.
Diagonal elements of A306364 in even-numbered rows.

Programs

  • Magma
    I:=[1, 3]; [n le 2 select I[n] else (4*n^2-8*n+3)*Self(n-1): n in [1..20]]; // Vincenzo Librandi, Nov 18 2014
  • Maple
    a:= n-> (d-> d(2*n-1)*d(2*n+1))(doublefactorial):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 30 2013
# second Maple program:
A079484 := n-> LinearAlgebra[Determinant](Matrix(2*n+1, (i, j)-> `if`(i+j=2*n+1, j, i))): seq(A079484(n), n=0..14); # Rainer Rosenthal, Jun 18 2024
  • Mathematica
    a[n_] := (2n - 1)!!*(2n + 1)!!; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Jan 30 2013 *)
  • PARI
    /* Formula using the zeta function and a log integral:*/
L(n)= intnum(t=0, 1, log(1-1/t)^n);
Zetai(n)= -I*I^n*(2*Pi)^(n-1)/(n-1)*L(1-n);
a(m)={my(n=m+1);round(real(-I*2^(2*n-1)*Zetai(1/2-n)*L(-1/2+n)/(Zetai(-1/2+n)*L(1/2-n))))};
/* Gerry Martens, Mar 07 2011, adapted to offset 0 by Hugo Pfoertner, Jun 19 2024 */
  • PARI
    {a(n) = if( n<0, -1 / self()(-1-n), (2*n + 1)! * (2*n)! / (n! * 2^n)^2 )}; /* Michael Somos, May 04 2017 */
  • PARI
    {a(n) = if( n<0, -1 / self()(-1-n), my(m = 2*n + 1); m! * polcoeff( x / sqrt( 1 - x^2 + x * O(x^m) ), m))}; /* Michael Somos, May 04 2017 */
  • PARI
    \\ using the Pochhammer symbol
a(n) = {my(P(x,k)=gamma(x+k)/gamma(x)); 4^n*round(P(1/2,n)*P(3/2,n))} \\ Hugo Pfoertner, Jun 20 2024
  • PARI
    \\ Scott's (1881) method
a(n) = {my(m=2*n+1, X = polroots(x^m-1), Y = polroots(x^m+1), M = matrix(m, m, i, j, 1/(X[i]-Y[j]))); (-1)^n * round(2^m * real(matpermanent(M)))}; \\ Hugo Pfoertner, Jun 23 2024

Formula

D-finite with recurrence a(n) = (4*n^2 - 1) * a(n-1) for all n in Z.
a(n) = A001147(n)*A001147(n+1).
E.g.f.: 1/(1-x^2)^(3/2) (with interpolated zeros). - Paul Barry, May 26 2003
a(n) = (2n+1)! * C(2n, n) / 2^(2n). - Ralf Stephan, Mar 22 2004.
Alternatingly signed values have e.g.f. sqrt(1+x^2).
a(n) is the value of the n-th moment of (1/Pi)*BesselK(1, sqrt(x)) on the positive part of the real line. - Olivier Gérard, May 20 2009
a(n) = -2^(2*n-1)*exp(i*n*Pi)*gamma(1/2+n)/gamma(3/2-n). - Gerry Martens, Mar 07 2011
E.g.f. (odd powers) tan(arcsin(x)) = Sum_{n>=0} (2n-1)!!*(2n+1)!!*x^(2*n+1)/(2*n+1)!. - Vladimir Kruchinin, Apr 22 2011
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - ((2*k+2)^2-1)/(1-x/(x - 1/G(k+1))); ( continued fraction ). - Sergei N. Gladkovskii, Jan 15 2013
a(n) = (2^(2*n+3)*Gamma(n+3/2)*Gamma(n+5/2))/Pi. - Jean-François Alcover, Jul 20 2015
Limit_{n->oo} 4^n*(n!)^2/a(n) = Pi/2. - Daniel Suteu, Feb 05 2017
From Michael Somos, May 04 2017: (Start)
a(n) = (2*n + 1) * A001818(n).
E.g.f.: Sum_{n>=0} a(n) * x^(2*n+1) / (2*n+1)! = x / sqrt(1 - x^2) = tan(arcsin(x)).
Given e.g.f. A(x) = y, then x * y' = y + y^3.
a(n) = -1 / a(-1-n) for all n in Z.
0 = +a(n)*(+288*a(n+2) -60*a(n+3) +a(n+4)) +a(n+1)*(-36*a(n+2) -4*a(n+3)) +a(n+2)*(+3*a(n+2)) for all n in Z. (End)
a(n) = Sum_{k=0..2n} (k+1) * A316728(n,k). - Alois P. Heinz, Jul 12 2018
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + L_1(1)*Pi/2, where L is the modified Struve function.
Sum_{n>=0} (-1)^n/a(n) = 1 - H_1(1)*Pi/2, where H is the Struve function. (End)

Extensions

Simpler description from Daniel Flath (deflath(AT)yahoo.com), Mar 05 2004
Showing 1-2 of 2 results.

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