\\ A355042: a(n) is the number of minimal balanced subsets of a set of n labeled elements.
\\ The following program is suitable for terms a(1)..a(5) and a(6) taking about 10 minutes.

\\ Checks a collection of sets to see if it is minimal and balanced.
\\ The collection is represented as a matrix M. Each column of the matrix
\\ gives an element of the collection in binary form.
\\ Returns 0,1,-1: 
\\   1 means M is a solution.
\\   0 means M is not a solution but can still be a subset of one.
\\  -1 means M is not a subset of any solution.
ChkMatrix(M)={
  if(matrank(M) < #M, -1,
      my( Y = vectorv(matsize(M)[1],i,1), \\ all 1's column vector
          X = matsolve(M, Y) );
      if(M*X-Y, 0, if(#select(t->t<=0, X), -1, 1));
  )
}

a(n)={
  \\ array of all bit sets as column vectors. These will be used to build the matrices.
  my(S=vector(2^n-1, k, my(t=binary(k)); concat(vector(n-#t), t)~));
  \\ recursive function to build the sets and test matrices.
  my(recurse(u,k) = 
    my(f=if(#u, ChkMatrix(Mat(u)))); 
    if(f, f>0, sum(k=1, k, self()(concat(u,[S[k]]), k-1))));
  \\ call the recursion
  recurse([], #S); 
}

\\ show the first 5 values
vector(5, n, a(n))