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

0,6

Comments

Row sums are A001147 (Double factorial).

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

Links

Table of n, a(n) for n=0..72.

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

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

Adjacent sequences: A067307 A067308 A067309 * A067311 A067312 A067313

Keyword

nonn,tabl

Author

Henry Bottomley, Jan 14 2002

Status

approved