cp's OEIS Frontend

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).

%I A261070 #45 Dec 12 2024 23:31:35
%S A261070 1,1,2,1,4,4,2,4,9,15,15,31,24,35,44,20,50
%N 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).
%C A261070 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 A261070 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 A261070 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 A261070 A250001(n) = Sum_{k>=0} T(n,k).
%F A261070 A000081(n+1) = T(n,0).
%e A261070 n\k 0  1  2  3  4  5  6
%e A261070 0   1
%e A261070 1   1
%e A261070 2   2  1
%e A261070 3   4  4  2  4
%e A261070 4   9 15 15 31 24 35 44
%e A261070 5  20 50  .  .  .  .  .  .  .  .  .
%Y A261070 Row sums give A250001.
%Y A261070 Cf. A000081, A152947, A249752, A252158, A280786 (column k=1)
%K A261070 nonn,more,tabf
%O A261070 0,3
%A A261070 _Benoit Jubin_, Aug 08 2015
%E A261070 T(4,2)..T(5,0) (6 terms) from _Travis Vasquez_, Nov 28 2024