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A280787 Triangle read by rows: number of topologically distinct sets of n circles with one pair intersecting, by number of factors.

%I A280787 #14 Apr 18 2019 11:51:21
%S A280787 1,3,1,10,4,1,30,15,4,1,91,50,16,4,1,268,162,55,16,4,1,790,506,185,56,
%T A280787 16,4,1,2308,1558,594,190,56,16,4,1,6737,4727,1878,617,191,56,16,4,1,
%U A280787 19609,14227,5825,1970,622,191,56,16,4,1
%N A280787 Triangle read by rows: number of topologically distinct sets of n circles with one pair intersecting, by number of factors.
%H A280787 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.
%e A280787 Triangle begins:
%e A280787      1;
%e A280787      3,    1;
%e A280787     10,    4,   1;
%e A280787     30,   15,   4,   1;
%e A280787     91,   50,  16,   4,  1;
%e A280787    268,  162,  55,  16,  4,  1;
%e A280787    790,  506, 185,  56, 16,  4, 1;
%e A280787   2308, 1558, 594, 190, 56, 16, 4, 1;
%e A280787 ...
%t A280787 a81[n_] := a81[n] = If[n <= 1, n, Sum[a81[n - j]*DivisorSum[j, #1*a81[#1] &], {j, n - 1}]/(n - 1)];
%t A280787 A027852[n_] := Module[{dh = 0, np}, For[np = 0, np <= n, np++, dh = a81[np]*a81[n - np] + dh]; If[EvenQ[n], dh = a81[n/2] + dh]; dh/2];
%t A280787 A280788[n_] := If[n == 0, 1, Sum[a81[np+1]*A027852[n-np+2], {np, 0, n}]];
%t A280787 t[n_] := t[n] = Module[{d, j}, If[n == 1, 1, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n - j], {j, 1, n - 1}]/(n - 1)]];
%t A280787 b[1, 1, 1] = 1;
%t A280787 b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[t[i] + j - 1, j], {j, 0, Min[n/i, p]}]]]]; A033185[n_, k_] := b[n, n, k];
%t A280787 A280786[n_] := If[n < 2, 0, Sum[A280787[n, f], {f, 1, n - 1}]];
%t A280787 A280787[n_, f_] := A280787[n, f] = Module[{ct}, Which[f == n, Return[0], f == n - 1, Return[1], f == 1, Return[A280786[n - 1] + A280788[n - 2]], True, ct = 0; Do[ct += A280787[np, 1]*A033185[n - np, f - 1], {np, 1, n - 1}]]; ct];
%t A280787 Table[A280787[n, f], {n, 2, 11}, {f, 1, n - 1}] // Flatten (* _Jean-François Alcover_, Nov 23 2017, after _R. J. Mathar_ and _Alois P. Heinz_ *)
%Y A280787 Row sums give A280786.
%K A280787 nonn,tabl
%O A280787 2,2
%A A280787 _N. J. A. Sloane_, Jan 20 2017