N-lower central series

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


 * The first member of the series is $$G$$, i.e., $$H_1 = G$$.
 * For any $$n$$, $$H_{n+1}$$ is the subgroup of $$G$$ generated by all n-commutators $$[x,y]_n = (xy)^ny^{-n}x^{-n}$$ where $$x \in H_n, y \in G$$.