  It is well known that weakly $p$-summable sequences in a Banach space $E$ are
associated to bounded operators from $\ell_{p^*}$ to $E$, and unconditionally
$p$-summable sequences in $E$ are associated to compact operators from
$\ell_{p^*}$ to $E$. Generalizing these results to a quite wide environment, we
characterize the classes of Banach spaces-valued sequences that are associated
to (or represented by) some Banach operator ideal. Using these
characterizations, we decide, among all sequence classes that usually appear in
the literature, which are represented by some Banach operator ideal and which
are not. Moreover, to each class that is represented by some Banach operator
ideal, we show an ideal that represents it. Illustrative examples and
additional applications are provided.
