  Most of known multipartite Bell inequalities involve correlation functions
for all subsystems. They are useless for entangled states without such
correlations. We give a method of derivation of families of Bell inequalities
for N parties, which involve, e.g., only (N-1)-partite correlations, but still
are able to detect proper N-partite entanglement. We present an inequality
which reveals five-partite entanglement despite only four-partite correlations.
Classes of inequalities introduced here can be put into a handy form of a
single non-linear inequality. An example is given of an N qubit state, which
strongly violates such an inequality, despite having no N-qubit correlations.
This surprising property might be of potential value for quantum information
tasks.
