  Hankel determinants of sequences related to Bernoulli and Euler numbers have
been studied before, and numerous identities are known. However, when a
sequence is shifted by one unit, the situation often changes significantly. In
this paper we use classical orthogonal polynomials and related methods to prove
a general result concerning Hankel determinants for shifted sequences. We then
apply this result to obtain new Hankel determinant evaluations for a total of
$13$ sequences related to Bernoulli and Euler numbers, one of which concerns
Euler polynomials.
