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A330053 Number of non-isomorphic set-systems of weight n with at least one singleton.

%I A330053 #18 Feb 10 2020 06:52:48
%S A330053 0,1,1,3,6,14,32,79,193,499,1321,3626,10275,30126,91062,284093,912866,
%T A330053 3018825,10261530,35814255,128197595,470146011,1764737593,6773539331,
%U A330053 26561971320,106330997834,434195908353,1807306022645,7663255717310,33079998762373
%N A330053 Number of non-isomorphic set-systems of weight n with at least one singleton.
%C A330053 A set-system is a finite set of finite nonempty sets of positive integers. An singleton is an edge of size 1. The weight of a set-system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%H A330053 Jean-François Alcover, <a href="/A330053/b330053.txt">Table of n, a(n) for n = 0..50</a> [using data from A283877 and A306005]
%F A330053 a(n) = A283877(n) - A306005(n). - _Jean-François Alcover_, Feb 09 2020
%e A330053 Non-isomorphic representatives of the a(1) = 1 through a(5) = 14 multiset partitions:
%e A330053   {1}  {1}{2}  {1}{12}    {1}{123}      {1}{1234}
%e A330053                {1}{23}    {1}{234}      {1}{2345}
%e A330053                {1}{2}{3}  {1}{2}{12}    {1}{12}{13}
%e A330053                           {1}{2}{13}    {1}{12}{23}
%e A330053                           {1}{2}{34}    {1}{12}{34}
%e A330053                           {1}{2}{3}{4}  {1}{2}{123}
%e A330053                                         {1}{2}{134}
%e A330053                                         {1}{2}{345}
%e A330053                                         {1}{23}{45}
%e A330053                                         {2}{13}{14}
%e A330053                                         {1}{2}{3}{12}
%e A330053                                         {1}{2}{3}{14}
%e A330053                                         {1}{2}{3}{45}
%e A330053                                         {1}{2}{3}{4}{5}
%t A330053 A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {_, _}][[All, 2]]];
%t A330053 A283877 = A@283877;
%t A330053 A306005 = A@306005;
%t A330053 a[n_] := A283877[[n + 1]] - A306005[[n + 1]];
%t A330053 a /@ Range[0, 50] (* _Jean-François Alcover_, Feb 09 2020 *)
%Y A330053 The complement is counted by A306005.
%Y A330053 The multiset partition version is A330058.
%Y A330053 Non-isomorphic set-systems with at least one endpoint are A330052.
%Y A330053 Non-isomorphic set-systems counted by vertices are A000612.
%Y A330053 Non-isomorphic set-systems counted by weight are A283877.
%Y A330053 Cf. A007716, A055621, A302545, A317533, A317794, A319559, A320665, A330055, A330056, A330057.
%K A330053 nonn
%O A330053 0,4
%A A330053 _Gus Wiseman_, Nov 30 2019