A fragment with the same provably recursive functions as n iterated inductive definitions is obtained by restricting second order arithmetic in the following way. The underlying language allows only up to n+1 nested second order quantifications and those are in such a way, that no second order variable occurs free in the scope of another second order quantifier. The amount of induction on arithmetical formulae only affects the arithmetical consequences of these theories, whereas adding induction for arbitrary formulae increases the strength by one inductive definition.
"Induction and inductive definitions in fragments of second order arithmetic." J. Symbolic Logic 70 (4) 1087 - 1107, December 2005. https://doi.org/10.2178/jsl/1129642116