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A269800 Convolution of A000107 and A027852.

%I A269800 #15 Dec 16 2018 03:29:50
%S A269800 0,0,1,3,10,30,91,268,790,2308,6737,19609,57044,165796,481823,1400028,
%T A269800 4068577,11825459,34380152,99981942,290854486,846397344,2463892294,
%U A269800 7174933683,20900764811,60904875999,177535250815,517673673674,1509950058629,4405547856394,12857716906991
%N A269800 Convolution of A000107 and A027852.
%C A269800 This counts the arrangements of n nested circles in the plane where one pair of circles touches. a(2)=1 because the (only) pair must touch. a(3)=3 because either the third circle circumscribes the touching pair or is inside one of the touching circles or is entirely separated from the touching pair.
%H A269800 Vincenzo Librandi, <a href="/A269800/b269800.txt">Table of n, a(n) for n = 0..500</a>
%H A269800 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), row sums Table 8.
%t A269800 b[0] = 0; b[1] = 1; b[n_] := b[n] =Sum[Sum[d b[d], {d, Divisors[j]}] b[n - j], {j, 1, n - 1}]/(n - 1);
%t A269800 a7[n_] := a7[n] = b[n] + Sum[ a7[n - i] b[i], {i, 1, n - 1}];
%t A269800 c[n_] := c[n] = If[n <= 1, n, (Sum[Sum[d c[d], {d, Divisors[j]}] c[n - j], {j, 1, n - 1}])/(n - 1)];
%t A269800 a52[n_] := (Sum[c[i] c[n-i], {i, 0, n}] + If[Mod[n, 2] == 0, c[n/2], 0])/2;
%t A269800 a[n_] := Sum[a7[k] a52[n - k + 1], {k, 0, n + 1}];
%t A269800 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Dec 16 2018, after _Alois P. Heinz_ in A000107 and A027852 *)
%Y A269800 Cf. A000107, A027852.
%K A269800 nonn
%O A269800 0,4
%A A269800 _R. J. Mathar_, Mar 05 2016