Codimensions of varieties of Poisson algebras with Lie nilpotent commutants

We study varieties of Poisson algebras defined by the identities $\{x_1,x_2\}\cdot\{x_3,x_4\}=0$ and $\{\{x_1,x_2\},\ldots,\{x_{2s+1}, x_{2s+2}\}\}=0$, $s\geq 1$. For each of the varieties we find a carrier algebra and build a basis of the $n$th proper polylinear space. We derive exact formulas for exponential generating functions for sequences of codimensions and proper codimensions as well as exact formulas for codimensions and proper codimensions.