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A067310
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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).
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6
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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; internal format)
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OFFSET
| 0,6
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LINKS
| H. Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311
alt.math.recreational discussion
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FORMULA
| Sum_{0<=j<n} (-1)^j * C((n-j)*(n-j+1)/2-1-i, n-1) * (C(2n, j)-C(2n, j-1))
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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.
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CROSSREFS
| Row sums are A001147 (Double factorial). Columns include A000108 (Catalan) for k=0 and A002694 for k=1. A067311 has a different view of the same table.
Sequence in context: A109077 A137585 A072458 * A122890 A138497 A113129
Adjacent sequences: A067307 A067308 A067309 * A067311 A067312 A067313
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KEYWORD
| nonn,tabl
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Jan 14 2002
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