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%I A055621 #26 Oct 14 2022 05:12:51
%S A055621 1,1,4,34,1952,18664632,12813206150470528,
%T A055621 33758171486592987151274638874693632,
%U A055621 1435913805026242504952006868879460423801146743462225386100617731367239680
%N A055621 Number of covers of an unlabeled n-set.
%D A055621 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 78 (2.3.39)
%H A055621 Alois P. Heinz, <a href="/A055621/b055621.txt">Table of n, a(n) for n = 0..12</a>
%H A055621 Heller, Jürgen <a href="https://doi.org/10.1016/j.jmp.2016.07.008">Identifiability in probabilistic knowledge structures</a>.  J. Math. Psychol. 77, 46-57 (2017).
%H A055621 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cover.html">Cover</a>
%F A055621 a(n) = (A003180(n) - A003180(n-1))/2 = A000612(n) - A000612(n-1) for n>0.
%F A055621 Euler transform of A323819. - _Gus Wiseman_, Aug 14 2019
%e A055621 There are 4 nonisomorphic covers of {1,2}, namely {{1},{2}}, {{1,2}}, {{1},{1,2}} and {{1},{2},{1,2}}.
%e A055621 From _Gus Wiseman_, Aug 14 2019: (Start)
%e A055621 Non-isomorphic representatives of the a(3) = 34 covers:
%e A055621   {123}  {1}{23}    {1}{2}{3}      {1}{2}{3}{23}
%e A055621          {13}{23}   {1}{3}{23}     {1}{2}{13}{23}
%e A055621          {3}{123}   {2}{13}{23}    {1}{2}{3}{123}
%e A055621          {23}{123}  {2}{3}{123}    {2}{3}{13}{23}
%e A055621                     {3}{13}{23}    {1}{3}{23}{123}
%e A055621                     {12}{13}{23}   {2}{3}{23}{123}
%e A055621                     {1}{23}{123}   {3}{12}{13}{23}
%e A055621                     {3}{23}{123}   {2}{13}{23}{123}
%e A055621                     {13}{23}{123}  {3}{13}{23}{123}
%e A055621                                    {12}{13}{23}{123}
%e A055621 .
%e A055621   {1}{2}{3}{13}{23}     {1}{2}{3}{12}{13}{23}    {1}{2}{3}{12}{13}{23}{123}
%e A055621   {1}{2}{3}{23}{123}    {1}{2}{3}{13}{23}{123}
%e A055621   {2}{3}{12}{13}{23}    {2}{3}{12}{13}{23}{123}
%e A055621   {1}{2}{13}{23}{123}
%e A055621   {2}{3}{13}{23}{123}
%e A055621   {3}{12}{13}{23}{123}
%e A055621 (End)
%p A055621 b:= proc(n, i, l) `if`(n=0, 2^(w-> add(mul(2^igcd(t, l[h]),
%p A055621       h=1..nops(l)), t=1..w)/w)(ilcm(l[])), `if`(i<1, 0,
%p A055621       add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i)))
%p A055621     end:
%p A055621 a:= n-> `if`(n=0, 2, b(n$2, [])-b(n-1$2, []))/2:
%p A055621 seq(a(n), n=0..8);  # _Alois P. Heinz_, Aug 14 2019
%t A055621 b[n_, i_, l_] := b[n, i, l] = If[n==0, 2^Function[w, Sum[Product[2^GCD[t, l[[h]]], {h, 1, Length[l]}], {t, 1, w}]/w][If[l=={}, 1, LCM@@l]], If[i<1, 0, Sum[b[n-i*j, i-1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]]];
%t A055621 a[n_] := If[n==0, 2, b[n, n, {}] - b[n-1, n-1, {}]]/2;
%t A055621 a /@ Range[0, 8] (* _Jean-François Alcover_, Jan 31 2020, after _Alois P. Heinz_ *)
%Y A055621 Unlabeled set-systems are A000612 (partial sums).
%Y A055621 The version with empty edges allowed is A003181.
%Y A055621 The labeled version is A003465.
%Y A055621 The T_0 case is A319637.
%Y A055621 The connected case is A323819.
%Y A055621 The T_1 case is A326974.
%Y A055621 Cf. A058891, A319559, A326946, A326973.
%K A055621 easy,nonn
%O A055621 0,3
%A A055621 _Vladeta Jovovic_, Jun 04 2000
%E A055621 More terms from David Moews (dmoews(AT)xraysgi.ims.uconn.edu) Jul 04 2002
%E A055621 a(0) = 1 prepended by _Gus Wiseman_, Aug 14 2019

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