A280786 Number of topologically distinct sets of n circles with one pair intersecting.

1, 4, 15, 50, 162, 506, 1558, 4727, 14227, 42521, 126506, 374969, 1108476, 3269902, 9630631, 28328999, 83251569, 244471484, 717486860, 2104777227, 6172357873, 18096097750, 53044095421, 155464365080, 455601800970, 1335107222743, 3912330438784, 11464463809180, 33595343643160

Offset: 2

Author

N. J. A. Sloane, Jan 20 2017

Keywords

• nonn

Links

• R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016, row sums Table 7.

Crossrefs

Row sums of A280787.

Column k=1 of A261070.

Programs

Maple

A280786 := proc(N)
    if N < 2 then
        0;
    else
        add(A280787(N,f),f=1..N-1) ;
    end if;
end proc:
A280787 := proc(N,f)
    option remember ;
    local Npr,ct ;
    if f = N then
        return 0;
    elif f = N-1 then
        return 1;
    elif f = 1 then
        A280786(N-1)+A280788(N-2) ;
    else
        ct := 0 ;
        for Npr from 1 to N-1 do
            ct := ct+procname(Npr,1)*A033185(N-Npr,f-1) ;
        end do:
        ct ;
    end if;
end proc:
seq(A280786(n),n=2..30) ; # R. J. Mathar, Mar 06 2017

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[A280786[n], {n, 2, 30}] (* Jean-François Alcover, Nov 23 2017, after R. J. Mathar and Alois P. Heinz *)