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Doppler-based planet surveys have discovered numerous giant planets but areincomplete beyond several AU. At larger star-planet separations, direct planetdetection through high-contrast imaging has proven successful, but thistechnique is sensitive only to young planets and characterization relies upontheoretical evolution models. Here we demonstrate that radial velocitymeasurements and high-contrast imaging can be combined to overcome theseissues. The presence of widely separated companions can be deduced byidentifying an acceleration (long-term trend) in the radial velocity of a star.By obtaining high spatial resolution follow-up imaging observations, we ruleout scenarios in which such accelerations are caused by stellar binarycompanions with high statistical confidence. We report results from an analysisof Doppler measurements of a sample of 111 M-dwarf stars with a median of 29radial velocity observations over a median time baseline of 11.8 yr. Bytargeting stars that exhibit a radial velocity acceleration ("trend") withadaptive optics imaging, we determine that 6.5% +/- 3.0% of M-dwarf stars hostone or more massive companions with 1 < m/M_Jupiter < 13 and 0 < a < 20 AU.These results are lower than analyses of the planet occurrence rate aroundhigher-mass stars. We find the giant planet occurrence rate is described by adouble power law in stellar mass M and metallicity F = [Fe/H] such that f(M,F)= 0.039(+0.056,-0.028) M^(0.8(+1.1,-0.9)) 10^((3.8 +/- 1.2)F). Our results areconsistent with gravitational microlensing measurements of the planetoccurrence rate; this study represents the first model-independent comparisonwith microlensing observations.

The TRENDS High-Contrast Imaging Survey. IV. The Occurrence Rate of Giant Planets around M-Dwarfs