A simple method for evaluating partition functions of linear systems is presented. The method produces the generating function of the partition functions of systems of all sizes. The method keeps the simplicity of Lifson's sequence-generating function method. Unlike the sequence-generating function method, which only applies to infinite systems, the method presented here applies to both finite and infinite systems. Explicit expression of the partition functions can be readily derived directly from the generating function. The method provides a simple protocol to handle a large range of models that come out in studying cooperative phenomena in physical chemistry and biophysics.