Descartes' rule of signs, Rolle's theorem and sequences of admissible pairs

Given a real univariate degree \(d\) polynomial \(P\), the numbers \(pos_k\) and \(neg_k\) of positive and negative roots of \(P^{(k)}\), \(k=0\), \(\ldots\), \(d-1\), must be admissible, i.e. they must satisfy certain inequalities resulting from Rolle's theorem and from Descartes' rule of signs. For \(1\leq d\leq 5\), we give the answer to the question for which admissible \(d\)-tuples of pairs \((pos_k\), \(neg_k)\) there exist polynomials \(P\) with all nonvanishing coefficients such that for \(k=0\), \(\ldots\), \(d-1\), \(P^{(k)}\) has exactly \(pos_k\) positive and \(neg_k\) negative roots all of which are simple.