  The aim of this paper is to compute the first homology of
$\operatorname{IA}_n$ with coefficients in the $\operatorname{Aut}(F_n)$-module
$A_2(n)$ of Jacobi diagrams of degree $2$ on $n$-component oriented arcs. We
also prove the self-duality of the $\operatorname{Aut}(F_3)$-submodule
$A''_2(3)$ of $A_2(3)$ which is generated by a certain element.
