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%I #37 Dec 16 2021 04:11:15
%S 1,1,2,1,4,4,2,4,9,15
%N 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).
%C 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).
%C 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
%H R. J. Mathar, <a href="http://arxiv.org/abs/1603.00077">Topologically Distinct Sets of Non-intersecting Circles in the Plane</a>, arXiv:1603.00077 [math.CO], 2016.
%F A250001(n) = Sum_{k>=0} T(n,k).
%F A000081(n+1) = T(n,0).
%e n\k 0 1 2 3 4 5 6
%e 0 1
%e 1 1
%e 2 2 1
%e 3 4 4 2 4
%e 4 9 15 . . . . .
%e 5 20 .
%Y Row sums give A250001.
%Y Cf. A000081, A249752, A252158, A280786 (column k=1)
%K nonn,more,tabf
%O 0,3
%A _Benoit Jubin_, Aug 08 2015
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