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%I A055152 #17 Oct 14 2022 05:13:43
%S A055152 0,1,14,956,9331320,6406603065901952,
%T A055152 16879085743296493569230716352778240,
%U A055152 717956902513121252476003434439730211883694285987816199468264943161704448
%N A055152 Proper covers of an unlabeled n-set.
%H A055152 Alois P. Heinz, <a href="/A055152/b055152.txt">Table of n, a(n) for n = 1..12</a>
%H A055152 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 A055152 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ProperCover.html">Proper covers</a>
%F A055152 a(n) = (A003180(n) - 2*A003180(n-1))/4.
%F A055152 Apparently a(n) = A002857(n) - A000612(n-1). - _R. J. Mathar_, Apr 22 2007
%p A055152 b:= proc(n, i, l) `if`(n=0, 2^(w-> add(mul(2^igcd(t, l[h]),
%p A055152       h=1..nops(l)), t=1..w)/w)(ilcm(l[])), `if`(i<1, 0,
%p A055152       add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i)))
%p A055152     end:
%p A055152 a:= n->  (b(n$2, [])-2*b(n-1$2, []))/4:
%p A055152 seq(a(n), n=1..8);  # _Alois P. Heinz_, Aug 14 2019
%t A055152 b[n_] := Sum[1/Function[p, Product[Function[c, j^c*c!][Coefficient[p, x, j]], {j, 1, Exponent[p, x]}]][Total[x^l]]*2^(Function[w, Sum[Product[ 2^GCD[t, l[[i]]], {i, 1, Length[l]}], {t, 1, w}]/w][If[l == {}, 1, LCM @@ l]]), {l, IntegerPartitions[n]}];
%t A055152 a[n_] := (b[n] - 2 b[n - 1])/4;
%t A055152 a /@ Range[8] (* _Jean-François Alcover_, Feb 19 2020, after _Alois P. Heinz_ in A000612 *)
%Y A055152 See A007537 for labeled case. Cf. A055621.
%K A055152 easy,nonn
%O A055152 1,3
%A A055152 _Vladeta Jovovic_, Jun 14 2000
%E A055152 More terms from _David Wasserman_, Mar 21 2002

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