cp's OEIS Frontend

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

%I A280786 #17 Apr 18 2019 11:51:12
%S A280786 1,4,15,50,162,506,1558,4727,14227,42521,126506,374969,1108476,
%T A280786 3269902,9630631,28328999,83251569,244471484,717486860,2104777227,
%U A280786 6172357873,18096097750,53044095421,155464365080,455601800970,1335107222743,3912330438784,11464463809180,33595343643160
%N A280786 Number of topologically distinct sets of n circles with one pair intersecting.
%H A280786 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 7.
%p A280786 A280786 := proc(N)
%p A280786     if N < 2 then
%p A280786         0;
%p A280786     else
%p A280786         add(A280787(N,f),f=1..N-1) ;
%p A280786     end if;
%p A280786 end proc:
%p A280786 A280787 := proc(N,f)
%p A280786     option remember ;
%p A280786     local Npr,ct ;
%p A280786     if f = N then
%p A280786         return 0;
%p A280786     elif f = N-1 then
%p A280786         return 1;
%p A280786     elif f = 1 then
%p A280786         A280786(N-1)+A280788(N-2) ;
%p A280786     else
%p A280786         ct := 0 ;
%p A280786         for Npr from 1 to N-1 do
%p A280786             ct := ct+procname(Npr,1)*A033185(N-Npr,f-1) ;
%p A280786         end do:
%p A280786         ct ;
%p A280786     end if;
%p A280786 end proc:
%p A280786 seq(A280786(n),n=2..30) ; # _R. J. Mathar_, Mar 06 2017
%t A280786 a81[n_] := a81[n] = If[n <= 1, n, Sum[a81[n - j]*DivisorSum[j, #1*a81[#1] &], {j, n - 1}]/(n - 1)];
%t A280786 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 A280786 A280788[n_] := If[n == 0, 1, Sum[a81[np + 1]*A027852[n - np + 2], {np, 0, n}]];
%t A280786 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 A280786 b[1, 1, 1] = 1;
%t A280786 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 A280786 A280786[n_] := If[n < 2, 0, Sum[A280787[n, f], {f, 1, n - 1}]];
%t A280786 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 A280786 Table[A280786[n], {n, 2, 30}] (* _Jean-François Alcover_, Nov 23 2017, after _R. J. Mathar_ and _Alois P. Heinz_ *)
%Y A280786 Row sums of A280787.
%Y A280786 Column k=1 of A261070.
%K A280786 nonn
%O A280786 2,2
%A A280786 _N. J. A. Sloane_, Jan 20 2017