  We study scalar products of Bethe vectors in the models solvable by the
nested algebraic Bethe ansatz and described by $\mathfrak{gl}(m|n)$
superalgebra. Using coproduct properties of the Bethe vectors we obtain a sum
formula for their scalar products. This formula describes the scalar product in
terms of a sum over partitions of Bethe parameters. We also obtain recursions
for the Bethe vectors. This allows us to find recursions for the highest
coefficient of the scalar product.
