  We introduce a class of recursions defined over the $d$-dimensional integer
lattice. The discrete equations we study are interpreted as higher dimensional
extensions to the discrete Toda lattice equation. We shall prove that the
equations satisfy the coprimeness property, which is one of integrability
detectors analogous to the singularity confinement test. While the degree of
their iterates grows exponentially, their singularities exhibit a nature
similar to that of integrable systems in terms of the coprimeness property. We
also prove that the equations can be expressed as mutations of a seed in the
sense of the Laurent phenomenon algebra.
