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A290867 Irregular triangle read by rows: the number of points that are the intersections of k semicircles in the configuration A290447(n).

%I A290867 #19 Mar 07 2020 11:26:23
%S A290867 0,0,0,0,1,0,5,0,15,0,35,0,70,0,123,1,0,195,5,0,285,15,0,420,25,0,586,
%T A290867 39,2,0,818,53,4,0,1110,73,6,0,1451,103,10,0,1846,142,18,0,2361,181,
%U A290867 26,0,2956,234,33,2,0,3704,287,40,4,0,4567,348,49,8
%N A290867 Irregular triangle read by rows: the number of points that are the intersections of k semicircles in the configuration A290447(n).
%C A290867 Row lengths are A290726(n).
%C A290867 The first entry of each row is 0, because an intersection requires at least 2 lines.
%C A290867 The first row with 3 entries is for n=9, because that is the first configuration with a nontrivial intersection.
%C A290867 Row sums give A290447.
%H A290867 David Applegate, <a href="/A290867/b290867.txt">Table of n, a(n) for n = 1..800</a>
%H A290867 David Applegate, <a href="/A290867/a290867.txt">Triangular table T(n,k) for n = 1..100</a>
%H A290867 N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, <a href="https://vimeo.com/237029685">Part I</a>, <a href="https://vimeo.com/237030304">Part 2</a>, <a href="https://oeis.org/A290447/a290447_slides.pdf">Slides.</a> (Mentions this sequence)
%H A290867 N. J. A. Sloane (in collaboration with Scott R. Shannon), <a href="/A331452/a331452.pdf">Art and Sequences</a>, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
%F A290867 Sum_{k} T(n,k) * binomial(k,2) = binomial(n,4), because there are binomial(n,4) total pairs of semicircles, and an intersection of k consists of binomial(k,2) of those pairs.
%F A290867 A290865(n) = binomial(n,2) + Sum_{k} T(n,k) * (k-1).
%e A290867 Triangle begins:
%e A290867   0;
%e A290867   0;
%e A290867   0;
%e A290867   0,   1;
%e A290867   0,   5;
%e A290867   0,  15;
%e A290867   0,  35;
%e A290867   0,  70;
%e A290867   0, 123,   1;
%e A290867   0, 195,   5;
%e A290867   0, 285,  15;
%e A290867   0, 420,  25;
%e A290867   0, 586,  39,   2;
%Y A290867 Cf. A290447, A290726, A290865, A290866.
%K A290867 nonn,tabf
%O A290867 1,7
%A A290867 _David Applegate_, Aug 12 2017