The equivalence relation isologism partitions the class of all groups into families, and it is a well-known fact that the Baer invariant is a powerful tool for this classification. In this article, we provide an explicit formula for the Baer invariant of a free $n$th nilpotent group (the $n$th nilpotent product of infinite cyclic groups, $\\\\\\\\\\\\\\\\textbf{ Z}\\\\\\\\\\\\\\\\stackrel{n}* \\\\\\\\\\\\\\\\textbf{ Z}\\\\\\\\\\\\\\\\stackrel{n}*\\\\\\\\\\\\\\\\ldots \\\\\\\\\\\\\\\\stackrel{n}*\\\\\\\\\\\\\\\\textbf{ Z}$) with respect to the variety of polynilpotent groups of class row $(c_1,c_2)$ (which is called polynilpotent multiplier), when $(c_2+1)n-(c_2+1)<c_1$ and $c_2<5$. The conclusion actually generalizes [16].

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