|
| |
|
|
A067310
|
|
Square table read by antidiagonals of number of ways of arranging n chords on a circle with k simple intersections (i.e., no intersections with 3 or more chords).
|
|
6
|
|
|
|
1, 0, 1, 0, 0, 2, 0, 0, 1, 5, 0, 0, 0, 6, 14, 0, 0, 0, 3, 28, 42, 0, 0, 0, 1, 28, 120, 132, 0, 0, 0, 0, 20, 180, 495, 429, 0, 0, 0, 0, 10, 195, 990, 2002, 1430, 0, 0, 0, 0, 4, 165, 1430, 5005, 8008, 4862, 0, 0, 0, 0, 1, 117, 1650, 9009, 24024, 31824, 16796, 0, 0, 0, 0, 0, 70, 1617
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,6
|
|
|
COMMENTS
|
Row sums are A001147 (Double factorial).
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n,k) = Sum_{j=0..n-1} (-1)^j * C((n-j)*(n-j+1)/2-1-k, n-1) * (C(2n, j) - C(2n, j-1)) where C(r,s)=binomial(r,s) if r>=s>=0 and 0 otherwise.
|
|
|
EXAMPLE
|
Rows start:
1, 0, 0, 0, 0, 0, 0, ...;
1, 0, 0, 0, 0, 0, 0, ...;
2, 1, 0, 0, 0, 0, 0, ...;
5, 6, 3, 1, 0, 0, 0, ...;
14, 28, 28, 20, 10, 4, 1, ...; etc.,
i.e., there are 5 ways of arranging 3 chords with no intersections, 6 with one, 3 with two and 1 with three.
|
|
|
CROSSREFS
|
A067311 has a different view of the same table.
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
