  We consider the lower-triangular matrix of generating polynomials that
enumerate $k$-component forests of rooted trees on the vertex set $[n]$
according to the number of improper edges (generalizations of the Ramanujan
polynomials). We show that this matrix is coefficientwise totally positive and
that the sequence of its row-generating polynomials is coefficientwise
Hankel-totally positive. More generally, we define the generic rooted-forest
polynomials by introducing also a weight $m! \, \phi_m$ for each vertex with
$m$ proper children. We show that if the weight sequence $\phi$ is
Toeplitz-totally positive, then the two foregoing total-positivity results
continue to hold. Our proofs use production matrices and exponential Riordan
arrays.
