  The modified B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r},
\quad n > 0 introduced by D. Zagier in 1998 are extended to the polynomial case
by replacing $B_{r}$ by the Bernoulli polynomials $B_{r}(x)$. Properties of
these new polynomials are established using the umbral method as well as
classical techniques. The values of $x$ that yield periodic subsequences
$B_{2n+1}^{*}(x)$ are classified. The strange 6-periodicity of $B_{2n+1}^{*}$,
established by Zagier, is explained by exhibiting a decomposition of this
sequence as the sum of two parts with periods 2 and 3, respectively. Similar
results for modifications of Euler numbers are stated.
