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).

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

Author

Henry Bottomley, Jan 14 2002

Keywords

• nonn
• tabl

Comments

Row sums are A001147 (Double factorial).

Columns include A000108 (Catalan) for k=0 and A002694 for k=1.

Examples

			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.
		

Links

• H. Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311
• J. Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25, g_n(x).
• alt.math.recreational discussion

Crossrefs

A067311 has a different view of the same table.

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.