Hopf's formula for nilpotent multiplier

Statement
Let $$G$$ be a group isomorphic to the quotient group $$F/R$$, where $$F$$ is a free group and $$R$$ is a normal subgroup of $$F$$. Suppose $$c$$ is a positive integer. Then, the $$c$$- nilpotent multiplier of $$G$$, denoted $$M^{(c)}(G)$$, is an abelian group given by the formula:

$$M^{(c)}(G) \cong (R \cap \gamma_{c+1}(F))/\gamma_{c+1}(R,F)$$.

Here:

Note that any choice of generating set for $$G$$ gives a choice of $$F$$ and $$R$$ for which the theorem can be applied: $$F$$ is the free group on those generators with the natural surjection, and $$R$$ is the kernel of the surjection.
 * $$\gamma_{c+1}(F)$$ is the $$(c + 1)^{th}$$ member of the lower central series of $$F$$, which is described explicitly as a $$(c+1)$$-fold iterated commutator of copie of $$F$$. Inductively, it is defined as $$\gamma_1(F) = F$$ and $$\gamma_{i+1}(F) = [F,\gamma_i(F)]$$
 * $$\gamma_{c+1}(R,F)$$ is defined as the $$(c+1)^{th}$$ member of the series defined inductively as $$\gamma_1(R,F) = R$$ and $$\gamma_{i+1}(R,F) = [F,\gamma_i(R,F)]$$

Related facts

 * Hopf's formula for Baer invariant
 * Hopf's formula for Schur multiplier: This is the special case $$c = 1$$.

Applications

 * Formula for nilpotent multiplier of free nilpotent group