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A002866 a(0) = 1; for n>0, a(n) = 2^(n-1)*n!.
(Formerly M3604 N1463)
51
1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960, 92897280, 1857945600, 40874803200, 980995276800, 25505877196800, 714164561510400, 21424936845312000, 685597979049984000, 23310331287699456000, 839171926357180416000, 31888533201572855808000, 1275541328062914232320000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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, 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, Jan 27 2003

This is also the denominator of the integral of ((1-x^2)^(n-1/2))/(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 nonempty lists. - Bob Proctor, Apr 18 2005

Row sums of A131222. - Paul Barry, 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, Nov 28 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, Jul 25 2008

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 churchgoers into pews and then linearly order the nonempty pews. - Geoffrey Critzer, Mar 16 2009

Equals the row sums of A156992. - Geoffrey Critzer, Mar 05 2010

From Gary W. Adamson, May 17 2010: (Start)

Next term in the series = (1, 3, 5, 7, ...) dot (1, 1, 4, 24, ...);

e.g., a(5) = 1920 = (1, 3, 5, 7, 9) dot (1, 1, 4, 24, 192) = (1 + 3 + 20 + 168 + 1728). (End)

a(n) is the number of ways to represent the permutations of {1,2,...,n} in cycle notation, taking into account that we can permute the order of all cycles and also have k ways to write a length-k cycle.

For positive n, a(n) equals the permanent of the n X n matrix with consecutive integers 1 to n along the main diagonal, consecutive integers 2 to n along the subdiagonal, and 1's everywhere else. - John M. Campbell, Jul 10 2011

From Dennis P. Walsh, Nov 26 2011: (Start)

Number of ways to arrange n books on consecutive bookshelves.

To derive a(n) = n!2^(n-1), we note that there are n! ways to arrange the books in a row. Then there are 2^(n-1) ways to place the arranged books on consecutive shelves since there are 2^(n-1) ordered partitions of n. Hence a(n) = n!2^(n-1).

Also, a(n) is the number of ways to stack n different alphabet blocks in contiguous stacks.

Furthermore, a(n) is the number of labeled, rooted forests that have (i) each root labeled larger than any nonroot, (ii) each root having exactly one child node, (iii) n non-root nodes, and (iv) each node in the forest with at most one child node.

Example: a(3)=24 since there are 24 arrangements of books b1, b2, and b3 on consecutive shelves, namely, |b1 b2 b3|, |b1 b3 b2|, |b2 b1 b3|, |b2 b3 b1|, |b3 b1 b2|, |b3 b2 b1|, |b1 b2||b3|, |b2 b1| |b3|, |b1 b3||b2|, |b3 b1||b2|, |b2 b3||b1|, |b3 b2||b1|, |b1||b2 b3|,|b1||b3 b2|, |b2||b1 b3|, |b2||b3 b1|, |b3||b1 b2|, |b3||b2 b1|, |b1||b2||b3|, |b1||b3||b2|, |b2||b1||b3|, |b2||b3||b1|, |b3||b1||b2|, and |b3||b2||b1|.

(End)

For n > 3, a(n) is the order of the Coxeter group (also called Weyl group) of type D_n. - Tom Edgar, Nov 05 2013

REFERENCES

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

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)

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

T. D. Noe, Table of n, a(n) for n = 0..100

J.-P. Bultel, A. Chouria, J.-G. Luque and O. Mallet, Word symmetric functions and the Redfield-Polya theorem, Accepted to FPSAC 2013, 2013. <hal-00793788>.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

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

Joël Gay, Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.

W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 121

Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.

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

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

OEIS Wiki, Sorting numbers

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

Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math.CO/0606404 [math.CO], Jan 05 2007.

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

Index to divisibility sequences

Index entries for sequences related to shoe lacings

Index entries for related partition-counting sequences

FORMULA

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

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

For n>=1, a(n) = Sum_{i=0..m/2} (-1)^i * binomial(n, i) * (n-2*i)^n. - Yong Kong (ykong(AT)curagen.com), Dec 28 2000

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. is B(A(x)) where B(x)=1/(1-x) and A(x)=x/(1-x). - Geoffrey Critzer, Mar 16 2009

a(n) = Sum_{k=1..n} A156992(n,k). - Dennis P. Walsh, Nov 26 2011

a(n+1) = Sum_{k=0..n} A132393(n,k)*2^(n+k), n>0. - Philippe Deléham, Nov 28 2011

G.f.: 1 + x/(1-4x/(1-2x/(1-6x/(1-4x/(1-8x/(1-6x/(1-10x/(1-... (continued fraction). - Philippe Deléham, Nov 29 2011

a(n) = 2*n*a(n-1) for n>=2. - Dennis P. Walsh, Nov 29 2011

G.f.: (1 + 1/G(0))/2  where G(k)=  1 +2*x*k - 2*x*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 02 2012

G.f.: 1 + x/Q(0),m=4, where Q(k) = 1 - m*x*(2*k+1) - m*x^2*(2*k+1)*(2*k+2)/( 1 - m*x*(2*k+2) - m*x^2*(2*k+2)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 23 2013

G.f.: 1 + x/(G(0) -x), where G(k) = 1 + x*(k+1) - 4*x*(k+1)/(1 - x*(k+2)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 24 2013

a(n) = Sum_{k=0..n} L(n,k)*k!; L(n,k) the unsigned Lah numbers. - Peter Luschny, Oct 18 2014

a(n) = round(Sum_{k>=1} log(k)^n/k^(3/2))/4,  for n>=1, which is related to the n-th derivative of zeta(x) evaluated at x=3/2. - Richard R. Forberg, Jan 02 2015

a(n) = n!*hypergeom([-n+1],[],-1)) for n>=1. - Peter Luschny, Apr 08 2015

EXAMPLE

For the shoe lacing: with the notation introduced in A078602 the a(3-1)=4 "straight" lacings for 3 pairs of eyelets 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 centers 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 centers 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.

MAPLE

A002866 := n-> `if`(n=0, 1, 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, Oct 16 2006

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); # Zerinvary Lajos, Apr 04 2009

MATHEMATICA

s=1; lst={1, s}; Do[s+=(s*=n); AppendTo[lst, s], {n, 2, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)

Join[{1}, Table[2^(n-1) n!, {n, 20}]] (* Harvey P. Dale, Sep 27 2013 *)

PROG

(FORTRAN) See Pfoertner link.

(PARI) a(n)=if(n, n!<<(n-1), 1) \\ Charles R Greathouse IV, Jan 13 2012

(PARI) a(n) = if(n == 0, 1, 2^(n-1)*n!);

vector(100, n, a(n-1)) \\ Altug Alkan, Oct 18 2015

CROSSREFS

Cf. A028371, A078602, A078698, A078702, A156992.

Bisections give A002671 and A274304.

Appears in A167584 (n=>1); equals the row sums of A167594 (n=>1). - Johannes W. Meijer, Nov 12 2009

Sequence in context: A259868 A036691 A293021 * A182798 A073840 A024249

Adjacent sequences:  A002863 A002864 A002865 * A002867 A002868 A002869

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 20 02:57 EST 2018. Contains 317371 sequences. (Running on oeis4.)