OFFSET
0,1
COMMENTS
The collection C may include the empty set and/or U.
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.
LINKS
Andrew Snowden, On the representation theory of the symmetry group of the Cantor set, arXiv:2308.06648 [math.RT], 2023.
FORMULA
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.
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
MATHEMATICA
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;
a /@ Range[0, 8] (* Jean-François Alcover, Jul 20 2011, after formula *)
PROG
(PARI) C(n) = sum(k=0, n, binomial(n, k)*(-1)^(n-k)*2^(2^k)); \\ A000371
a(n) = 0^n - 1^n + sum(k=0, n, binomial(n, k)*(-1)^(n-k)*C(k)); \\ Michel Marcus, Oct 27 2020
(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
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
David Pasino, Sep 29 2007
STATUS
approved