Galois groups of Taylor polynomials: prescriptions and arithmetic families

Let $f_n(x)=\sum_{j=0}^n a_jx^j$ be the Taylor polynomials of a power series with nonzero rational coefficients. We study the proportions of degrees for which the Galois group over \Q is symmetric, alternating, or neither. We prove that the set of degrees $n\geq3$ with symmetric group can be chosen arbitrarily, even for an entire function with positive rational coefficients whose Taylor polynomials have distinct negative real roots. More precisely, at each degree $n\geq3$ outside the chosen set, $f_n$ has exactly one rational root, and the group on the other roots is the full symmetric group. Thus the proportions need not converge; when they do, the proportion of other groups can be any number in $[0,1]$. These conclusions answer negatively two questions of Katz and Rivin. Moreover, for fixed s\in\Q\setminus\Z, we also give a condition on the prime divisors of rational multipliers of the coefficients $s/(s+j)$ that forces symmetric groups in density one. For the Taylor polynomials of $(1-x)^{-s}$, we determine the groups in all sufficiently large degrees. The number of exceptions to the symmetric group up to degree $X$ is $c_s\sqrt X+O_s(1)$, with an explicit constant $c_s\geq0$. We also determine the groups of the central-binomial Taylor polynomials in every degree and prove that every nonconstant Catalan Taylor polynomial has symmetric group. This proves Sun's Catalan group conjecture and his irreducible-reduction criterion for the monic reciprocals of the central-binomial polynomials.