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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
OFFSET
0,6
COMMENTS
Row sums are A001147 (Double factorial).
Columns include A000108 (Catalan) for k=0 and A002694 for k=1.
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.
Sequence in context: A361290 A285638 A325667 * A369321 A122890 A309524
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Jan 14 2002
STATUS
approved