N-derived series

Definition
Suppose $$G$$ is a group and $$n$$ is an integer. The $$n$$-derived series of $$G$$ is defined as the following descending series $$H_n, n \in \mathbb{N}_0$$:


 * The zeroth member of the series is $$G$$, i.e., $$H_0 = G$$.
 * For any $$n \ge 0$$, $$H_{n+1}$$ is the n-derived subgroup of $$H_n$$, i.e., it is the subgroup of $$G$$ generated by all n-commutators $$[x,y]_n = (xy)^ny^{-n}x^{-n}$$ where $$x,y \in H_n$$.