%I A001710 M2933 N1179
%S A001710 1,1,1,3,12,60,360,2520,20160,181440,1814400,19958400,239500800,
%T A001710 3113510400,43589145600,653837184000,10461394944000,177843714048000,
%U A001710 3201186852864000,60822550204416000,1216451004088320000
%N A001710 Order of alternating group A_n, or number of even permutations of n letters.
%C A001710 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
%C A001710 a(n) is the number of Hamiltonian circuit masks for an n X n adjacency 
               matrix of an undirected graph. - Chad R. Brewbaker (crb002(AT)iastate.edu), 
               Jan 31 2003
%C A001710 a(n) 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 
               (crb002(AT)iastate.edu), Jan 31 2003
%C A001710 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 (deutsch(AT)duke.poly.edu), 
               Aug 28 2004
%C A001710 Minimum permanent over all n X n (0,1)-matrices with exactly n/2 zeros. 
               - Simone Severini (ss54(AT)york.ac.uk), Oct 15 2004
%C A001710 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,
               ... .
%C A001710 Starting (1, 3, 12, 60,...) = binomial transform of A000153: (1, 2, 7, 
               32, 181,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 25 
               2008]
%C A001710 First column of A092582. [From Mats Granvik (mats.granvik(AT)abo.fi), 
               Feb 08 2009]
%C A001710 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 
               2009: (Start)
%C A001710 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.
%C A001710 (End)
%D A001710 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001710 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001710 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.
%D A001710 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               88.
%D A001710 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.
%H A001710 N. J. A. Sloane, <a href="b001710.txt">Table of n, a(n) for n = 0..100</
               a>
%H A001710 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A001710 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas 
               for Some Functions on Finite Sets</a>
%H A001710 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=262">
               Encyclopedia of Combinatorial Structures 262</a>
%H A001710 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               On generalizations of Stirling number triangles</a>, J. Integer Seqs., 
               Vol. 3 (2000), #00.2.4.
%H A001710 Xah Lee, <a href="http://xahlee.org/MathGraphicsGallery_dir/Combinatorics_dir/
               loopNPoints.html">Combinatorics: Loop in n points</a>
%H A001710 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</
               a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A001710 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AlternatingGroup.html">Link to a section of The World of Mathematics.</
               a>
%H A001710 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CircularPermutation.html">Link to a section of The World of Mathematics.</
               a>
%H A001710 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HamiltonianCircuit.html">Link to a section of The World of Mathematics.</
               a>
%H A001710 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EvenPermutation.html">Even Permutation</a>
%H A001710 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               OddPermutation.html">Odd Permutation</a>
%H A001710 <a href="Sindx_Fa.html#factorial">Index entries for sequences related 
               to factorial numbers</a>
%F A001710 a(n) = numerator(n!/2) and A141044(n) = denominator(n!/2).
%F A001710 a(0) = a(1) = a(2) = 1; a(n)=n*a(n-1) for n>3. - Chad R. Brewbaker (crb002(AT)iastate.edu), 
               Jan 31 2003 [Corrected by N. J. A. Sloane (njas(AT)research.att.com), 
               Jul 25 2008]
%F A001710 a(0) = 0, a(1) = 1; a(n) = sum k*a(k) for k = 1 to n-1. - Amarnath Murthy 
               (amarnath_murthy(AT)yahoo.com), Oct 29 2002
%F A001710 Stirling transform of a(n+1)=[1, 3, 12, 160, ...] is A083410(n)=[1, 4, 
               22, 154, ...]. - Michael Somos Mar 04 2004
%F A001710 First Eulerian transform of A000027. See A000142 for definition of FET. 
               - Ross La Haye (rlahaye(AT)new.rr.com), Feb 14 2005
%F A001710 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 (pbarry(AT)wit.ie), Apr 18 2005
%F A001710 E.g.f.: (2-x^2)/(2-2*x). E.g.f. of a(n+2),n>=0, is 1/(1-x)^3.
%F A001710 a(n+1)= A136656(n,1)*(-1)^n, n>=1.
%p A001710 seq(mul((k), k=3..n), n=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Sep 14 2007
%p A001710 a[ -1]:=1:a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=(a[n-1]*(n+1)^2) 
               od: seq(sqrt(a[n]), n=-1..19); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 09 2008
%p A001710 with (combinat):seq(count(Partition((n!+1)), size=2), n=0..20); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2008
%t A001710 f[n_]:=If[n>1,n,1];a=2;lst={1};Do[a=n*a-a;AppendTo[lst,f[a/4]],{n,2,5!}];
               lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), May 28 2009]
%o A001710 (PARI) a(n)=if(n<2,n>=0,n!/2)
%Y A001710 Cf. A000142, A049444, A049459. a(n+1)= A046089(n, 1), n >= 1 (first column 
               of triangle).
%Y A001710 A000153 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 25 2008]
%Y A001710 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 
               2009: (Start)
%Y A001710 Cf. A161739 (q(n) sequence).
%Y A001710 (End)
%Y A001710 Sequence in context: A062569 A089057 A077134 this_sequence A105752 A053532 
               A159867
%Y A001710 Adjacent sequences: A001707 A001708 A001709 this_sequence A001711 A001712 
               A001713
%K A001710 nonn,easy,nice
%O A001710 0,4
%A A001710 N. J. A. Sloane (njas(AT)research.att.com).
%E A001710 More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001
%E A001710 Further from Simone Severini (ss54(AT)york.ac.uk), Oct 15 2004