A261070 Irregular triangle read by rows: T(n,k) is the number of arrangements of n circles with 2k intersections (using the same rules as A250001).

1, 1, 2, 1, 4, 4, 2, 4, 9, 15

Offset

0,3

Comments

Length of n-th row: 1 + (n-1)n/2 (for a configuration for T(n,(n-1)n/2), consider n circles of radius 1 and centers at (k/n,0) for 1<=k<=n).

The generating function down the column k=1 is 1+z^2 *C^2(z) *[C^2(z)+C(z^2)]/ (2*[1-z*C(z)]) = 1+ z^2 +4*z^3 +15*z^4+ 50*z^5+...where C(z) = 1+z+2*z^2+4*z^3+... is the g.f. of A000081 divided by z; eq. (78) in arXiv:1603.00077. - R. J. Mathar, Mar 05 2016

Links

Table of n, a(n) for n=0..9.

R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.

Formula

A250001(n) = Sum_{k>=0} T(n,k).

A000081(n+1) = T(n,0).

Example

n\k 0  1  2  3  4  5  6
0   1
1   1
2   2  1
3   4  4  2  4
4   9 15  .  .  .  .  .
5  20  .

Crossrefs

Row sums give A250001.

Cf. A000081, A249752, A252158, A280786 (column k=1)

Sequence in context: A128250 A086145 A309086 * A249140 A113421 A135366

Adjacent sequences: A261067 A261068 A261069 * A261071 A261072 A261073

Keyword

nonn,more,tabf

Author

Benoit Jubin, Aug 08 2015

Status

approved