|
|
A078698
|
|
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.
|
|
5
|
|
|
1, 2, 20, 396, 14976, 907200, 79315200, 9551001600, 1513528934400, 305106949324800, 76296489615360000, 23175289163980800000, 8404709419090575360000, 3587225703492542791680000, 1779970753996760560435200000, 1016036270188884847558656000000, 661106386935312429191528448000000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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,
|
|
REFERENCES
|
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 494.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
a(3) = 20: 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 lacings are: 124356 154326 153426 142536 145236 135246 and the following and their mirror images: 125346 124536 125436 152346 153246 152436 154236.
Examples for n=2,3,4 can be found following the FORTRAN program at given link.
|
|
MATHEMATICA
|
a[n_] := (n-1)!^2 Sum[Binomial[n-k, k]^2, {k, 0, n/2}];
|
|
PROG
|
FORTRAN program provided at Pfoertner link
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|