The representation of hydrogen-bonding contribution by Huang and Radosz or by Veytsman involves terms that require implicit computation when more than one type of bonds is present. To reduce the computational burden associated with the solution of a set of nonlinear implicit equations, a free energy expansion method was proposed for Veytsman statistics in the present study. Based on the normalized Veytsman statistics, the Helmholtz free energy for association was expanded around a reference value. The expanded free energy was compared with rigorous values and found to closely approximate the original term. The lattice-compatible expanded Veytsman term was then combined with the explicit non-random lattice-fluid model of present authors to give the explicit free energy expression for hydrogen-bonding non-random lattice-fluid. Parameters for the model are segment number and interaction energy for physical interaction, and association enthalpy and entropy for hydrogen bonding. For a binary mixture, a binary parameter is needed. To assure the consistency, thermodynamic properties such as pressure and chemical potential were derived from the expanded Helmholtz free energy. Calculated vapor-liquid equilibria using the expansion for alkane-alkanol mixtures were found to closely agree with experimental values and with rigorous calculation results.