I'm not sure whether it applies for this particular potential, but there are allowed invariant transformations in EAM that mean the function in the file is not necessarily exactly the same as the equation in the paper. It may be a mathematically equivalent formulation. This is especially true for alloy potentials. What really matters is whether the calculated properties (cohesive energies, transition energies, etc.) are the same, but the so-called effective pair format is sometimes useful for comparing EAM potentials.
More info can be found here:
• Y. Mishin, "Interatomic potentials for metals," in Handbook of Materials Modeling, edited by S. Yip (Springer, Dordrect, The Netherlands, 2005), Chap. 2.2, pp. 459-478.
• A.E. Carlsson, "Beyond pair potentials in elemental transition metals and semiconductors," in Solid State Physics, Volume 43, Ehrenreich and Turnbull (eds.) (1990).
Confusing, I know.