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, 15, 31, 24, 35, 44, 20, 50

Offset: 0

Author

Benoit Jubin, Aug 08 2015

Keywords

• nonn
• more
• tabf

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

Examples

			n\k 0  1  2  3  4  5  6
0   1
1   1
2   2  1
3   4  4  2  4
4   9 15 15 31 24 35 44
5  20 50  .  .  .  .  .  .  .  .  .
		

Links

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

Crossrefs

Row sums give A250001.

Cf. A000081, A152947, A249752, A252158, A280786 (column k=1)

Formula

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

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

Extensions

T(4,2)..T(5,0) (6 terms) from Travis Vasquez, Nov 28 2024