0,3

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

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

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

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

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

Row sums of A131222. - Paul Barry (pbarry(AT)wit.ie), Jun 18 2007

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

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

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]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

T. S. Motzkin, Sorting numbers ...: for a link to this paper see A000262.

N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6, p. 257.

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.

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)

T. D. Noe, 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 121

Hugo Pfoertner, Counting straight shoe lacings. FORTRAN program and results

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Index entries for sequences related to shoe lacings

Index entries for related partition-counting sequences

Greg Egan, HypercubeMathematical Details, revised Sunday, April 22, 2007.

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

E.g.f.: (1-x)/(1-2x) - Paul Barry (pbarry(AT)wit.ie), May 26 2003, corrected Jun 18 2007

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

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]

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.

a(3) = 24 because The 24 rotations of a three-dimensional cube fall into four distinct classes: (i) the identity, which leaves everything fixed;

(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;

(iii) 6 rotations which leave the centres of two edges fixed, comprising rotations of 180 degrees for each of 6 pairs of edges;

(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.

A002866 := n-> 2^(n-1)*n!;

with(combstruct); SeqSeqL := [S, {S=Sequence(U, card >= 1), U=Sequence(Z, card >=1)}, labeled];

seq(ceil(count(Subset(n))*count(Permutation(n))/2), n=0..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 16 2006

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]

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]

FORTRAN program to count shoe lacings available at Pfoertner link.

Cf. A002671, A028371.

Cf. A078602, A078698, A078702.

a(n) = n!*A011782(n).

Cf. A002671, A028371. Cf. A078602, A078698, A078702. a(n) = n!*A011782(n).

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 2009: (Start)

Appears in A167584 (n=>1).

Equals the row sums of A167594 (n=>1).

(End)

Sequence in context: A001506 A088815 A036691 this_sequence A073840 A024249 A007145

Adjacent sequences: A002863 A002864 A002865 this_sequence A002867 A002868 A002869

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).