A274819 Triangle read by rows: T(n,k) = total number of regions in all arrangements of n circles in the affine plane forming k regions, excluding the regions that do not belong to the circles.

1, 0, 4, 3, 0, 0, 12, 16, 10, 6, 21, 0, 0, 0

Offset: 1

Author

Omar E. Pol, Jul 07 2016

Keywords

• nonn
• tabf
• hard
• more

Comments

In other words: not counting the regions between circles.

Consider the arrangements of n circles described in A250001.

Examples

			Triangle begins:
1;
0, 4,  3;
0, 0, 12, 16, 10, 6, 21;
0, 0,  0, ...
...
For n = 3 and k = 5 there are 2 arrangements of 3 circles in the affine plane forming 5 regions, excluding the regions that do not belong to the circles, so T(3,5) = 2*5 = 10.
For n = 3 and k = 6 there is only one arrangement of 3 circles in the affine plane forming 6 regions, excluding the regions that do not belong to the circles, so T(3,6) = 1*6 = 6.
Of course, there is a right triangle of all zeros starting from the second row.
		

Crossrefs

Row sums give A274823(n).

First differs from A274818 at a(10).

Cf. A250001, A274777.

Formula

T(n,k) = k*A274777(n,k).