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

2,2

Links

Table of n, a(n) for n=2..56.

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

Example

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;
...

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 *)

Crossrefs

Row sums give A280786.

Sequence in context: A134285 A141811 A291538 * A126954 A176992 A319375

Adjacent sequences: A280784 A280785 A280786 * A280788 A280789 A280790

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jan 20 2017

Status

approved