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