cp's OEIS Frontend

A131288 a(n) = number of ways to choose a collection C of subsets of U = [1,2,...,n] such that Union_{S in C} = U, Intersection_{S in C} = empty set.

%I A131288 #50 Aug 16 2025 19:18:49
%S A131288 2,1,7,193,63775,4294321153,18446744022173838463,
%T A131288 340282366920938463205120190760593525761,
%U A131288 115792089237316195423570985008687907847825466794905548626109625623336235655679
%N A131288 a(n) = number of ways to choose a collection C of subsets of U = [1,2,...,n] such that Union_{S in C} = U, Intersection_{S in C} = empty set.
%C A131288 The collection C may include the empty set and/or U.
%C A131288 The number of covers of an n-set (A000371) is the inverse binomial transform of number of sets of subsets. The number of coverings with empty intersection is (to within a unit parity flutter and a fudge unit when n = 0) the inverse binomial transform of the number of coverings, i.e., the second inverse binomial transform of number of sets of subsets.
%C A131288 The number of supports of an n-qubit quantum register that are entangled for arbitrary probability amplitudes (the support of a quantum state is the set of basis kets associated with nonvanishing probability amplitudes). - _Szymon Lukaszyk_, Aug 12 2025
%H A131288 Szymon Łukaszyk, <a href="https://www.researchgate.net/publication/394356306_On_the_quantum_separability_of_qubit_registers">On the quantum separability of qubit registers</a>, 2025.
%H A131288 David Pasino, <a href="/A131288/a131288.pdf">Set Covers with Empty Intersection, and a few Related Counts</a>
%H A131288 Andrew Snowden, <a href="https://arxiv.org/abs/2308.06648">On the representation theory of the symmetry group of the Cantor set</a>, arXiv:2308.06648 [math.RT], 2023.
%F A131288 a(n) = -(-1)^n + Sum_{k=0..n} Sum_{t=0..n} binomial(n, k)*binomial(k, t)*(-1)^(n-t)*2^(2^t) for n > 0.
%F A131288 a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*2^k*(2^(2^(n-k))-1) for n > 0. - _Andrew Howroyd_, Oct 28 2020
%t A131288 a[n_] = (-1)^(n+1) + Sum[Binomial[n, k]*Binomial[k, t]*(-1)^(n-t)*2^(2^t), {k, 0, n}, {t, 0, k}]; a[0] = 2;
%t A131288 a /@ Range[0, 8] (* _Jean-François Alcover_, Jul 20 2011, after formula *)
%o A131288 (PARI) C(n) = sum(k=0, n, binomial(n, k)*(-1)^(n-k)*2^(2^k)); \\ A000371
%o A131288 a(n) = 0^n - 1^n + sum(k=0, n, binomial(n,k)*(-1)^(n-k)*C(k)); \\ _Michel Marcus_, Oct 27 2020
%o A131288 (PARI) a(n)={(n==0) + sum(k=0, n, (-1)^k*binomial(n,k)*2^k*(2^(2^(n-k))-1))} \\ _Andrew Howroyd_, Oct 28 2020
%Y A131288 Cf. A003465 (coverings by nonempty subsets), A000371 = 2 * A003465 (coverings allowing the empty set as one of the subsets).
%K A131288 nonn,nice
%O A131288 0,1
%A A131288 _David Pasino_, Sep 29 2007