Inerter-based vibration energy harvesters (VEHs) have been widely studied to harvest energy from large-scale structural vibrations. Recently, there have been efforts to increase the operation frequency bandwidth of VEHs by introducing a variety of stiffness and inertia nonlinearity. This paper proposes a new inerter-based VEH comprising an epicyclic-gearing inerter and a pendulum vibration absorber. The centrifugal force of the pendulum introduces a new type of inertia nonlinearity that broadens the frequency bandwidth. This inerter-pendulum VEH (IPVEH) is incorporated in a single-degree-of-freedom structure to demonstrate its performance and the equations of motion of the system are derived. The method of multiple scales is applied to derive the amplitude–frequency response relationship of the harvested power in the primary resonance. The harvested power is optimized through tuning the harvester’s electrical damping and the optimum power is benchmarked with that of conventional linear inerter-based VEHs. The results show that the IPVEH has larger bandwidth and harvested power and the improvement is correlated with the strength of its inertia nonlinearity.