It has become customary to prove binomial identities by means of the method for automated proofs as developed by Petkovšek, Wilf and Zeilberger [PWZ96]. In this paper, we wish to emphasize the role of ``human'' and constructive proofs in contrast with the somewhat lazy attitude of relaying on ``automated'' proofs. As a meaningful example, we consider the four formulas by Romik [Rom03], related to Motzkin and central trinomial numbers. We show that a proof of these identities can be obtained by using the method of coefficients, a human method only requiring hand computations.