%I A002866 M3604 N1463
%S A002866 1,1,4,24,192,1920,23040,322560,5160960,92897280,1857945600,
%T A002866 40874803200,980995276800,25505877196800,714164561510400,
%U A002866 21424936845312000,685597979049984000,23310331287699456000
%N A002866 a(0) = 1; for n>0, a(n) = 2^(n-1)*n!.
%C A002866 Right side of the binomial sum Sum( (-1)^i * binomial(n, i) * (n-2*i)^n, 
               i=0..n/2) = 2^(n-1) * n! - Yong Kong (ykong(AT)curagen.com), Dec 
               28 2000
%C A002866 Consider the set of n-1 odd numbers from 3 to 2n-1, i.e.{3, 5, ..2n-1}. 
               There are 2^(n-1) subsets from {} to {3, 5, 7, ..2n-1}; a(n) = the 
               sum of the products of terms of all the subsets. (Product for empty 
               set =1.) a(4) = 1+ 3 +5 +7 + 3*5 +3*7 + 5*7 + 3*5*7 = 192. - Amarnath 
               Murthy (amarnath_murthy(AT)yahoo.com), Sep 06 2002
%C A002866 Also a(n-1) gives the ways to lace a shoe that has n pairs of eyelets 
               such that there is a straight (horizontal) connection between all 
               adjacent eyelet pairs. - Hugo Pfoertner (hugo(AT)pfoertner.org), 
               Jan 27 2003
%C A002866 This is also the denominator of the integral of ((1-x^2)^(n-.5))/(pi/
               4) where x ranges from 0 to 1. The numerator is (2*x)!/(x!*2^x). 
               In both cases n starts at 1. E.g. the denominator when n=3 is 24 
               and the numerator is 15. - Al Hakanson (hawkuu(AT)excite.com), Oct 
               17 2003
%C A002866 Number of ways to use the elements of {1,..,n} once each to form a sequence 
               of non-empty lists. - Bob Proctor, Apr 18 2005
%C A002866 Row sums of A131222. - Paul Barry (pbarry(AT)wit.ie), Jun 18 2007
%C A002866 Number of rotational symmetries of an n-cube. The number of all symmetries 
               of an n-cube is A000165. See Egan for signed cycle notation, other 
               notes, tables and animation. - Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Nov 28 2007
%C A002866 The n-th term of this sequence is the number of regions into which n-dimensional 
               space is partitioned by the 2n hyperplanes of the form x_i=x_j and 
               x_i=-x_j (for 1 <= i < j <= n). - Edward Scheinerman (ers(AT)jhu.edu), 
               May 04 2008
%C A002866 a(n) is the number of ways to seat n church-goers into pews and then 
               linearly order the non-empty pews. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Mar 16 2009]
%D A002866 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002866 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002866 T. S. Motzkin, Sorting numbers ...: for a link to this paper see A000262.
%D A002866 N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6, p. 257.
%D A002866 T. S. Motzkin, Sorting numbers for cylinders and other classification 
               numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, 
               pp. 167-176.
%D A002866 A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", 
               Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, 
               Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.26)
%H A002866 T. D. Noe, <a href="b002866.txt">Table of n, a(n) for n=0..100</a>
%H A002866 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 A002866 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=121">
               Encyclopedia of Combinatorial Structures 121</a>
%H A002866 Hugo Pfoertner, <a href="http://www.randomwalk.de/shoelace/strlace.txt">
               Counting straight shoe lacings. FORTRAN program and results</a>
%H A002866 N. J. A. Sloane and Thomas Wieder, <a href="http://arXiv.org/abs/math.CO/
               0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 
               83-89.
%H A002866 <a href="Sindx_La.html#lacings">Index entries for sequences related to 
               shoe lacings</a>
%H A002866 <a href="Sindx_Par.html#partN">Index entries for related partition-counting 
               sequences</a>
%H A002866 Greg Egan, <a href="http://gregegan.customer.netspace.net.au/APPLETS/
               29/HypercubeNotes.html">HypercubeMathematical Details</a>, revised 
               Sunday, April 22, 2007.
%F A002866 a(n) ~ 2^(1/2)*pi^(1/2)*n^(3/2)*2^n*e^-n*n^n*{1 + 13/12*n^-1 + ...}. 
               - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
%F A002866 E.g.f.: (1-x)/(1-2x) - Paul Barry (pbarry(AT)wit.ie), May 26 2003, corrected 
               Jun 18 2007
%F A002866 1, 4, 24, 192, 1920,... is the exponential (or binomial) convolution 
               of 1,1,3,15,105,... and 1,3,15,105,945 (A001147). - David Callan 
               (callan(AT)stat.wisc.edu), Jul 25 2008
%F A002866 E.g.f. is B(A(x)) where B(x)=1/(1-x) and A(x)=x/(1-x) [From Geoffrey 
               Critzer (critzer.geoffrey(AT)usd443.org), Mar 16 2009]
%e A002866 For the shoe lacing: with the notation introduced in A078602 the a(3-1)=4 
               "straight" lacings for 3 pairs of eylets are: 125346, 125436, 134526, 
               143526. Their mirror images 134256, 143256, 152346, 152436 are not 
               counted.
%e A002866 a(3) = 24 because The 24 rotations of a three-dimensional cube fall into 
               four distinct classes: (i) the identity, which leaves everything 
               fixed;
%e A002866 (ii) 9 rotations which leave the centres of two faces fixed, comprising 
               rotations of 90, 180 and 270 degrees for each of 3 pairs of faces;
%e A002866 (iii) 6 rotations which leave the centres of two edges fixed, comprising 
               rotations of 180 degrees for each of 6 pairs of edges;
%e A002866 (iv) 8 rotations which leave two vertices fixed, comprising rotations 
               of 120 and 240 degrees for each of 4 pairs of vertices. For an n-cube, 
               rotations can be more complex. For example, in 4 dimensions a rotation 
               can either act in a single plane, such as the x-y plane, while leaving 
               any vectors orthogonal to that plane unchanged, or it can act in 
               two orthogonal planes, performing rotations in both and leaving no 
               vectors fixed. In higher dimensions, there will be room for more 
               planes and more choices as to the number of planes in which a given 
               rotation acts.
%p A002866 A002866 := n-> 2^(n-1)*n!;
%p A002866 with(combstruct); SeqSeqL := [S, {S=Sequence(U,card >= 1), U=Sequence(Z,
               card >=1)},labeled];
%p A002866 seq(ceil(count(Subset(n))*count(Permutation(n))/2),n=0..17); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Oct 16 2006
%p A002866 restart: G(x):=(1-x)/(1-2*x): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1],
               x) od:x:=0:seq(f[n],n=0..17);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 04 2009]
%t A002866 s=1;lst={1, s};Do[s+=(s*=n);AppendTo[lst, s], {n, 2, 5!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Nov 15 2008]
%o A002866 FORTRAN program to count shoe lacings available at Pfoertner link.
%Y A002866 Cf. A002671, A028371.
%Y A002866 Cf. A078602, A078698, A078702.
%Y A002866 a(n) = n!*A011782(n).
%Y A002866 Cf. A002671, A028371. Cf. A078602, A078698, A078702. a(n) = n!*A011782(n).
%Y A002866 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 
               2009: (Start)
%Y A002866 Appears in A167584 (n=>1).
%Y A002866 Equals the row sums of A167594 (n=>1).
%Y A002866 (End)
%Y A002866 Sequence in context: A001506 A088815 A036691 this_sequence A073840 A024249 
               A007145
%Y A002866 Adjacent sequences: A002863 A002864 A002865 this_sequence A002867 A002868 
               A002869
%K A002866 nonn,easy,nice
%O A002866 0,3
%A A002866 N. J. A. Sloane (njas(AT)research.att.com).