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.
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