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A001710
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Order of alternating group A_n, or number of even permutations of n letters.
(Formerly M2933 N1179)
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101
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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; internal format)
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OFFSET
| 0,4
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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 R. 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 R. 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
Comment from John Perry, Sep 20 2008: The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0,1,3,12,60,360,2520,20160,... .
Starting (1, 3, 12, 60,...) = binomial transform of A000153: (1, 2, 7, 32, 181,...). [From Gary W. Adamson, Dec 25 2008]
First column of A092582. [From 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). [From Reinhard Zumkeller, Feb 19 2010]
Contribution from Gary W. Adamson, Aug 02 2010: (Start)
Startting (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). (End)
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. [From Johannes W. Meijer, Apr 20 2011]
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REFERENCES
| 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.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 88.
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).
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.
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..100
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
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
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 numbers
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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>3. - Chad R. 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 (amarnath_murthy(AT)yahoo.com), 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))). [From 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). [From Wolfdieter Lang, Apr 30 2010]
a(n) = (n-2)! * t(n-1), n>1, where t(n)=A000217(n)..the n-th triangular number. [From Gary Detlefs, May 21 2010]
a(n) = ( A000254(n) - 2* A001711(n-3) )/3, n>2 [From Gary Detlefs, May 24 2010]
O.g.f.: 1 + x*Sum_{n>=0} n^n*x^n/(1 + n*x)^n. [From 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]
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MAPLE
| seq(mul(k, k=3..n), n=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 14 2007
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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 *)
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PROG
| (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 */
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CROSSREFS
| Cf. A000142, A049444, A049459. a(n+1)= A046089(n, 1), n >= 1 (first column of triangle), A000153, A161739 (q(n) sequence), A093468, A002260.
Sequence in context: A062569 A089057 A077134 * A105752 A177138 A053532
Adjacent sequences: A001707 A001708 A001709 * A001711 A001712 A001713
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001
Further from Simone Severini (simoseve(AT)gmail.com), Oct 15 2004
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