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We present an explicit structure for the Baer invariant of a free $n$thnilpotent group (the $n$th nilpotent product of infinite cyclic groups,$\textbf{Z}\st{n}* \textbf{Z}\st{n}*...\st{n}*\textbf{Z}$) with respect to thevariety ${\cal V}$ with the set of words $V=\{[\ga_{c_1+1},\ga_{c_2+1}]\}$, forall $c_1\geq c_2$ and $2c_2-c_1>2n-2$. Also, an explicit formula for thepolynilpotent multiplier of a free $n$th nilpotent group is given for any classrow $(c_1,c_2,...,c_t)$, where $c_1\geq n$.

Some Baer Invariants of Free Nilpotent Groups