  The Nekrasov--Okounkov hook length formula provides a fundamental link
between the theory of partitions and the coefficients of powers of the Dedekind
eta function. In this paper we examine three conjectures presented by
Amdeberhan. The first conjecture is a refined Nekrasov--Okounkov formula
involving hooks with trivial legs. We prove the conjecture. The second
conjecture is on properties of the roots of the underlying D'Arcais
polynomials. We give a counterexample and present a new conjecture. The third
conjecture is on the unimodality of the coefficients of the involved
polynomials. We confirm the conjecture up to the polynomial degree $1000$.
