  The modified Bernoulli numbers \begin{equation*} B_{n}^{*} = \sum_{r=0}^{n}
\binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 \end{equation*} introduced by D.
Zagier in 1998 were recently extended to the polynomial case by replacing
$B_{r}$ by the Bernoulli polynomials $B_{r}(x)$. Arithmetic properties of the
coefficients of these polynomials are established. In particular, the 2-adic
valuation of the modified Bernoulli numbers is determined. A variety of
analytic, umbral, and asymptotic methods is used to analyze these polynomials.
