A055152 Proper covers of an unlabeled n-set.

0, 1, 14, 956, 9331320, 6406603065901952, 16879085743296493569230716352778240, 717956902513121252476003434439730211883694285987816199468264943161704448

Offset: 1

Author

Vladeta Jovovic, Jun 14 2000

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..12
• Heller, Jürgen Identifiability in probabilistic knowledge structures. J. Math. Psychol. 77, 46-57 (2017).
• Eric Weisstein's World of Mathematics, Proper covers

Crossrefs

See A007537 for labeled case. Cf. A055621.

Programs

Maple

b:= proc(n, i, l) `if`(n=0, 2^(w-> add(mul(2^igcd(t, l[h]),
      h=1..nops(l)), t=1..w)/w)(ilcm(l[])), `if`(i<1, 0,
      add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i)))
    end:
a:= n->  (b(n$2, [])-2*b(n-1$2, []))/4:
seq(a(n), n=1..8);  # Alois P. Heinz, Aug 14 2019

Mathematica

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]}];
a[n_] := (b[n] - 2 b[n - 1])/4;
a /@ Range[8] (* Jean-François Alcover, Feb 19 2020, after Alois P. Heinz in A000612 *)

Formula

a(n) = (A003180(n) - 2*A003180(n-1))/4.

Apparently a(n) = A002857(n) - A000612(n-1). - R. J. Mathar, Apr 22 2007

Extensions

More terms from David Wasserman, Mar 21 2002