A078698 -id:A078698 - cp's OEIS frontend

A002866 a(0) = 1; for n > 0, a(n) = 2^(n-1)*n!.

1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960, 92897280, 1857945600, 40874803200, 980995276800, 25505877196800, 714164561510400, 21424936845312000, 685597979049984000, 23310331287699456000, 839171926357180416000, 31888533201572855808000, 1275541328062914232320000

Offset: 0

N. J. A. Sloane

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

			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.

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).

T. D. Noe, Table of n, a(n) for n = 0..100
Ofek Lauber Bonomo and Shlomi Reuveni, Occupancy correlations in the asymmetric simple inclusion process, arXiv:1905.02170 [cond-mat.stat-mech], 2019.
J.-P. Bultel and A. Chouria, J.-G. Luque and O. Mallet, Word symmetric functions and the Redfield-Polya theorem, Accepted to FPSAC 2013, 2013; arXiv:1302.5815 [math.CO], 2013.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. 3 (2000), #00.1.5.
Greg Egan, Hypercube Mathematical Details, 2007-2008.
Joël Gay and 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: 2013 45th Southeastern Symposium on System Theory (SSST).
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
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]
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, 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/0606404 [math.CO], 2007.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003.
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

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

FORTRAN
```
See Pfoertner link.
```

[1] cat [2^(n-1)*Factorial(n): n in [1..25]]; // G. C. Greubel, Jun 13 2019

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

Join[{1},Table[2^(n-1) n!,{n,25}]] (* Harvey P. Dale, Sep 27 2013 *)
a[n_] := (-1)^n Hypergeometric2F1Regularized[1, -n, 2 - n, 2];
Table[a[n], {n, 0, 20}]  (* Peter Luschny, Apr 26 2024 *)

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

a(n) = if(n == 0, 1, 2^(n-1)*n!);
vector(25, n, a(n-1)) \\ Altug Alkan, Oct 18 2015

[1] + [2^(n-1)*factorial(n) for n in (1..25)] # G. C. Greubel, Jun 13 2019

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 - 4*x/(1 - 2*x/(1 - 6*x/(1 - 4*x/(1 - 8*x/(1 - 6*x/(1 - 10*x/(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) are 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
From Amiram Eldar, Aug 04 2020: (Start)
Sum_{n >= 0} 1/a(n) = 2*sqrt(e) - 1.
Sum_{n >= 0} (-1)^n/a(n) = 2/sqrt(e) - 1. (End)

A051286 Whitney number of level n of the lattice of the ideals of the fence of order 2n.

Original entry on oeis.org

1, 1, 2, 5, 11, 26, 63, 153, 376, 931, 2317, 5794, 14545, 36631, 92512, 234205, 594169, 1510192, 3844787, 9802895, 25027296, 63972861, 163701327, 419316330, 1075049011, 2758543201, 7083830648, 18204064403, 46812088751, 120452857976

Offset: 0

Emanuele Munarini

A Chebyshev transform of the central trinomial numbers A002426: image of 1/sqrt(1-2x-3x^2) under the mapping that takes g(x) to (1/(1+x^2))*g(x/(1+x^2)). - Paul Barry, Jan 31 2005
a(n) has same parity as Fibonacci(n+1) = A000045(n+1); see A107597. - Paul D. Hanna, May 22 2005
This is the second kind of Whitney numbers, which count elements, not to be confused with the first kind, which sum Mobius functions. - Thomas Zaslavsky, May 07 2008
From Paul Barry, Mar 31 2010: (Start)
Apply the Riordan array (1/(1-x+x^2),x/(1-x+x^2)) to the aerated central binomial coefficients with g.f. 1/sqrt(1-4x^2).
Hankel transform is A174882. (End)
a(n) is the number of lattice paths in L[n]. The members of L[n] are lattice paths of weight n that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step h with weight 1, an (1,0)-step H with weight 2, a (1,1)-step U with weight 2, and a (1,-1)-step D with weight 1. The weight of a path is the sum of the weights of its steps. Example: a(3)=5 because we have hhh, hH, Hh, UD, and DU; a(4)=11 because we have hhhh, hhH, hHh, Hhh, HH, hUD, UhD, UDh, hDU, DhU, and DUh (see the Bona-Knopfmacher reference).
Apparently the number of peakless grand Motzkin paths of length n. - David Scambler, Jul 04 2013
A bijection between L[n] (as defined above) and peakless grand Motzkin paths of length n is now given in arXiv:2002.12874. - Sergi Elizalde, Jul 14 2021
a(n) is also the number of unimodal bargraphs with a centered maximum (i.e., whose column heights are weakly increasing in the left half and weakly decreasing in the right half) and semiperimeter n+1. - Sergi Elizalde, Jul 14 2021
Diagonal of the rational function 1 / ((1 - x)*(1 - y) - (x*y)^2). - Ilya Gutkovskiy, Apr 23 2025
a(n) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,2}, and no nodes have the same weight as their parent node. - John Tyler Rascoe, Jun 10 2025

			a(3) = 5 because the ideals of size 3 of the fence F(6) = { x1 < x2 > x3 < x4 > x5 < x6 } are x1*x3*x5, x1*x2*x3, x3*x4*x5, x1*x5*x6, x3*x5*x6.

Vincenzo Librandi and Alois P. Heinz, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Axel Bacher, Cyril Banderier and Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects, in International Conference on Language and Automata Theory and Applications, S. Klein, C. Martín-Vide, D. Shapira (eds), Springer, Cham, pp 195-206, 2018.
Andrei Asinowski, Axel Bacher, Cyril Banderier and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
Cyril Banderier and Paweł Hitczenko, Enumeration and asymptotics of restricted compositions having the same number of parts, Disc. Appl. Math. 160 (18) (2012) 2542-2554. See Puzzle 3.1.
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Grand Dyck paths with air pockets, arXiv:2211.04914 [math.CO], 2022.
Jean-Luc Baril and José L. Ramírez, Fibonacci and Catalan paths in a wall, 2023.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
Miklós Bóna and Arnold Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
Steffen Eger, The Combinatorics of Weighted Vector Compositions, arXiv:1704.04964 [math.CO], 2017.
Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2021.
Edyta Hetmaniok, Barbara Smoleń and Roman Wituła, The Stirling triangles, Proceedings of the Symposium for Young Scientists in Technology, Engineering and Mathematics (SYSTEM 2017), Kaunas, Lithuania, April 28, 2017, p. 35-41.
Ivo L. Hofacker, Christian M. Reidys, and Peter F. Stadler, Symmetric circular matchings and RNA folding. Discr. Math., 312:100-112, 2012. See Prop. 5, C_l^{1}(z).
Emanuele Munarini and Norma Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
Jesús Sistos Barrón and Hua Wang, Balanced n-color compositions, Integers (2024) Vol. 24A, Art. No. A2. See pp. 4, 5, 8, 14.

Cf. main diagonal of A125250, column k=2 of A384747.

Cf. A051291, A051292, A078698, A107597, A185828 (log), A174882 (Hankel transf.).

seq( sum('binomial(i-k,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1$2, 2, 5][n+1],
     ((2*n-1)*a(n-1)+(n-1)*a(n-2)+(2*n-3)*a(n-3)-(n-2)*a(n-4))/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 11 2016

Table[Sum[Binomial[n-k,k]^2,{k,0,Floor[n/2]}],{n,0,40}] (* Emanuele Munarini, Mar 01 2011; corrected by Harvey P. Dale, Sep 12 2012 *)
CoefficientList[Series[1/Sqrt[1-2*x-x^2-2*x^3+x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 05 2013 *)
a[n_] := HypergeometricPFQ[ {(1-n)/2, (1-n)/2, -n/2, -n/2}, {1, -n, -n}, 16]; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Feb 26 2013 *)

makelist(sum(binomial(n-k,k)^2,k,0,floor(n/2)),n,0,40);  /* Emanuele Munarini, Mar 01 2011 */

a(n)=polcoeff(1/sqrt((1+x+x^2)*(1-3*x+x^2)+x*O(x^n)),n)

a(n)=sum(k=0,n,binomial(n-k,k)^2) /* Paul D. Hanna */

{a(n)=polcoeff( exp(sum(m=1,n, sum(k=0,m, binomial(2*m,2*k)*x^k) *x^m/m) +x*O(x^n)), n)}  /* Paul D. Hanna, Mar 18 2011 */

{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

{a(n)=local(A=1+x); A=sum(m=0, n, x^m*sum(k=0, n, binomial(m+k, k)^2*x^k) * (1-x)^(2*m+1) +x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

{a(n)=local(A=1+x); A=sum(m=0, n\2, x^(2*m) * sum(k=0, n, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

{a(n)=local(A=1+x); A=sum(m=0, n\2, x^(2*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

from sympy import binomial
def a(n): return sum(binomial(n - k, k)**2 for k in range(n//2 + 1))
print([a(n) for n in range(31)]) # Indranil Ghosh, Apr 18 2017

G.f.: 1/sqrt(1 - 2*x - x^2 - 2*x^3 + x^4).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A002426(n-2k). - Paul Barry, Jan 31 2005
From Paul D. Hanna, May 22 2005: (Start)
a(n) = Sum_{k=0..n} C(n-k, k)^2.
Limit_{n->oo} a(n+1)/a(n) = (sqrt(5)+3)/2.
G.f.: 1/sqrt((1+x+x^2)*(1-3*x+x^2)). (End)
a(n) = Sum_{k=0..n} A049310(n, k)^2. - Philippe Deléham, Nov 21 2005
a(n) = Sum_{k=0..n} (C(k,k/2)*(1+(-1)^k)/2) * Sum_{j=0..n} (-1)^((n-j)/2)*C((n+j)/2,j)*((1+(-1)^(n-j))/2)*C(j,k). - Paul Barry, Mar 31 2010
G.f.: exp( Sum_{n>=1} (x^n/n)*Sum_{k=0..n} C(2n,2k)*x^k ). - Paul D. Hanna, Mar 18 2011
Logarithmic derivative equals A185828. - Paul D. Hanna, Mar 18 2011
D-finite with recurrence: n*a(n) - (2*n-1)*a(n-1) - (n-1)*a(n-2) - (2*n-3)*a(n-3) + (n-2)*a(n-4) = 0. - R. J. Mathar, Dec 17 2011
The g.f. A(x) satisfies the differential equation (1-2*x-x^2-2*x^3+x^4)*A'(x) = (1+x+3*x^2-2*x^3)*A(x), from which the recurrence conjectured by Mathar follows. - Emanuele Munarini, Dec 18 2017
a(n) ~ phi^(2*n + 2) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 05 2013, simplified Dec 18 2017
From Paul D. Hanna, Sep 05 2014: (Start)
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^k.
G.f.: Sum_{n>=0} x^n *[Sum_{k>=0} C(n+k,k)^2 * x^k] * (1-x)^(2*n+1).
G.f.: Sum_{n>=0} x^(2*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(2*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
(End)

A078700 Number of symmetric ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.

Original entry on oeis.org

1, 2, 6, 30, 192, 1560, 15120, 171360, 2217600, 32296320, 522547200, 9300614400, 180583603200, 3798482688000, 86044973414400, 2088355965696000, 54064489070592000, 1487129136869376000, 43312058119249920000

Offset: 1

Hugo Pfoertner, Dec 18 2002

nonn

The lace must pass through each eyelet exactly one and must begin and end at the extreme pair of eyelets.

			a(3) = 6: label the eyelets 1,2,3 from front to back on the left side then 4,5,6 from back to front on the right side. The symmetric lacings are: 124356 154326 153426 142536 145236 135246.

Index entries for sequences related to shoe lacings

Cf. A000045, A005922, A014417, A078676, A078698, A078702.

ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card <= 1)}, labelled]: seq(combstruct[count](ZL, size=n), n=0..18); # Zerinvary Lajos, Mar 26 2008

Table[Fibonacci[n + 1]*(n - 1)!, {n, 1, 19}] (* Zerinvary Lajos, Jul 09 2009 *)

(define (A078700 n) (* (A000142 (- n 1)) (A000045 (+ n 1)))) ;; Antti Karttunen, Jan 02 2007

a(n) = (n-1)!*Fibonacci(n+1) = A000142(n-1)*A000045(n+1). - Conjectured by Vladeta Jovovic, Jan 23 2005, proved by Antti Karttunen, Jan 06 2007.
Proof of Jovovic's conjecture from Antti Karttunen, Jan 06 2007: (Start)
Because of the symmetry and the beginning and ending conditions, we need to consider only n-1 eyelets on the other side. If considering only the distance from the starting and ending eyelets (the "level" of each eyelet-pair) through which the lace is traveling (but ignoring on which side it is), the lace will induce some permutation of {1..(n-1)} after it has visited half of the eyelets (and the remaining half of its route is wholly determined by the symmetry). This gives the factor (n-1)!. Independently of this, the condition that each eyelet has at least one direct connection to the opposite side, means that there is a simple bijection with binary strings of length n with no two consecutive 0's. I.e. we mark 0 when the lace stays on the same side and 1 when it crosses to the other side.
From the starting eyelet the lace can either cross to the other side (but not to the top one) or stay on the same side. However, after visiting half of the eylets, the lace MUST cross to the other side (on the same level), so this leaves n-1 eyelets where the choice is free, except that there can be no two consecutive 0's on the route. This gives the factor Fibonacci(n+1). (End)

More terms added by Antti Karttunen, Jan 02 2007

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.

cp's OEIS Frontend

A002866 a(0) = 1; for n > 0, a(n) = 2^(n-1)*n!.

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A051286 Whitney number of level n of the lattice of the ideals of the fence of order 2n.

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A078700 Number of symmetric ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.

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A078702 Number of ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.

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A072503 Number of ways to lace a shoe with n eyelet pairs such that there is no direct "horizontal" connection between any adjacent eyelet pair.

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A079410 Number of ways to lace a shoe that has n pairs of eyelets such that the lace does not cross itself between the eyelet rows.

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