  McNamara and Sagan conjectured that if $a_0,a_1, a_2, \ldots$ is a P\'olya
frequency (PF) sequence, then so is $a_0^2, a_1^2 -a_0a_2, a_2^2-a_1a_3,
\ldots$. We prove this conjecture for a natural class of PF-sequences which are
interpolated by polynomials. In particular, this proves that the columns of
Pascal's triangle are infinitely log-concave, as conjectured by McNamara and
Sagan. We also give counterexamples to the first mentioned conjecture.
  Our methods provide families of nonlinear operators that preserve the
property of having only real and non-positive zeros.
