  In this article we introduce a new type of Pascal pyramids. A regular squared
mosaic in the hyperbolic plane yields a $(h^2r)$-cube mosaic in space
$\mathbf{H}^2\!\times\!\mathbf{R}$ and the definition of the pyramid is based
on this regular mosaic. The levels of the pyramid inherit some properties from
the Euclidean and hyperbolic Pascal triangles. We give the growing method from
level to level and show some illustrating figures.
