cp's OEIS Frontend

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) is the number of 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

The lace is "directed": reversing the order of eyelets along the path counts as a different solution. It must begin and end at the extreme pair of eyelets,

The lace must follow a Hamiltonian path through the 2n eyelets and cannot pass in order though three adjacent eyelets that are in a line.

The lace is "directed": reversing the order of eyelets along the path counts as a different solution (cf. A078674).

The lace must follow a Hamiltonian path through the 2n eyelets. At least one of the neighbors of every eyelet must be on the other side of the shoe.

The lace is "undirected": reversing the order of eyelets along the path does not count as a different solution.

The lace must pass through each eyelet exactly once, must begin and end at the extreme pair of eyelets and cannot pass in order though three adjacent eyelets that are in a line.

The lace is "undirected": reversing the order of eyelets along the path does not count as a different solution.

The lace must pass through each eyelet exactly once, must begin and end at the extreme pair of eyelets and cannot pass in order though three adjacent eyelets that are in a line.

The lacing must not have any "straight connections" between adjacent eyelet pairs (e.g. 2<->2*n-1, 3<->2*n-2, 4<->2*n-3,....). There are no symmetric solutions.

From Sean A. Irvine, Oct 06 2024: (Start)

The lacing must begin and end with the top eyelet pair (1 and 2n in Pfoertner's numbering).

Every eyelet must be used.

Under versus over crossings are not considered distinct.

Three consecutive eyelets on the same side are not permitted.

Because there are no symmetric solutions, mirrored solutions can be handled by simply dividing the total solutions by 2. (End)

The lace must pass through each eyelet exactly once, must begin and end at the extreme pair of eyelets and each eyelet must have at least one direct connection to the opposite side. The corresponding sequence including all configs where the lace crosses itself in the space between the eyelet rows is A078698. The only symmetric crossing-free lacing is 1234 for N=2.