  We introduce the concept of s-Hankel hypermatrix, which generalizes both
Hankel matrices and generic hypermatrices. We study two determinantal ideals
associated to an s-Hankel hypermatrix: the ideal I<s,t> generated by certain 2
x 2 slice minors, and the ideal \tilde{I}<s,t> generated by certain 2 x 2
generalized minors. We describe the structure of these two ideals, with
particular attention to the primary decomposition of I<s,t>, and provide the
explicit list of minimal primes for large values of s. Finally we give some
geometrical interpretations and generalise a theorem of Watanabe.
