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

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, 16, 4, 1, 2308, 1558, 594, 190, 56, 16, 4, 1, 6737, 4727, 1878, 617, 191, 56, 16, 4, 1, 19609, 14227, 5825, 1970, 622, 191, 56, 16, 4, 1

Offset: 2

Author

N. J. A. Sloane, Jan 20 2017

Keywords

• nonn
• tabl

Examples

			Triangle begins:
     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, 16,  4, 1;
  2308, 1558, 594, 190, 56, 16, 4, 1;
...
		

Links

• R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.

Crossrefs

Row sums give A280786.

Programs

Mathematica

a81[n_] := a81[n] = If[n <= 1, n, Sum[a81[n - j]*DivisorSum[j, #1*a81[#1] &], {j, n - 1}]/(n - 1)];
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];
A280788[n_] := If[n == 0, 1, Sum[a81[np+1]*A027852[n-np+2], {np, 0, n}]];
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)]];
b[1, 1, 1] = 1;
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];
A280786[n_] := If[n < 2, 0, Sum[A280787[n, f], {f, 1, n - 1}]];
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];
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 *)