The dependence of the superconducting gaps in epitaxial Ba(Fe_{1-x}Co_{x})_2As_2 thin films on the nominal doping x (0.04 leq x leq 0.15) was studied by means of point-contact Andreev-reflection spectroscopy. The normalized conductance curves were well fitted by using the 2D Blonder-Tinkham-Klapwijk model with two nodeless, isotropic gaps -- although the possible presence of gap anisotropies cannot be completely excluded. The amplitudes of the two gaps Delta_{S} and Delta_{L} show similar monotonic trends as a function of the local critical temperature T_{c}^{A} (measured in the same point contacts) from 25 K down to 8 K. The dependence of the gaps on x is well correlated to the trend of the critical temperature, i.e. to the shape of the superconducting region in the phase diagram. When analyzed within a simple three-band Eliashberg model, this trend turns out to be compatible with a mechanism of superconducting coupling mediated by spin fluctuations, whose characteristic energy scales with T_{c} according to the empirical law Omega_{0}= 4.65*k_{B}*T_{c}, and with a total electron-boson coupling strength lambda_{tot}= 2.22 for x leq 0.10 (i.e. up to optimal doping) that slightly decreases to lambda_{tot}= 1.82 in the overdoped samples (x = 0.15).
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