This paper provides an alternative method to the topological recursion for computing the one-point function \(W_1(x)\) in the case of Laguerre and generalized Laguerre ensembles, i.e. an Hermitian matrix model with potential \(V(x)=x\) on \(\mathbb{R}_+\). The main result of the paper is to provide an explicit three terms recursion, which is similar to the one of J. Harer and D. Zagier [Invent. Math. 85, 457–485 (1986; Zbl 0616.14017)] or N. Do and P. Norbury [Topological recursion for irregular spectral curves, arXiv:1412.8334] in the Gaussian case, for computing the coefficients of the one-point function. The main advantage of this method compared to the topological recursion, is that it does not require the knowledge of the other correlation functions and therefore computations are much easier and faster. Thus, as explained in the paper, it is particularly well-suited for the matching of the coefficients with integers arising in enumerative geometry. In the Laguerre case, the author presents the details of the connection with the number of unicellular two-colored maps. The proof of the three terms recursion relies first on a replica trick, then the use of the Harish-Chandra-Itzykson-Zuber integral, and finally some contour deformations and standard complex analysis techniques. Consequently, the paper is relatively short and pleasant to read. However, since the proof relies on the knowledge of some specific formulas, the method may not easily generalize to more complicated models.