A bijective approach to the area of generalized Motzkin paths

For fixed positive integer k, let E_n denote the set of lattice paths using the steps (1, 1), (1, -1), and (k, 0) and running from (0, 0) to (n, 0) while remaining strictly above the x-axis elsewhere. We first prove bijectively that the total area of the regions bounded by the paths of E_n and the x-axis satisfies a four-term recurrence depending only on k. We then give both a bijective and a generating function argument proving that the total area under the paths of E_n equals the total number of lattice points on the x-axis hit by the unrestricted paths running from (0, 0) to (n - 2, 0) and using the same step set as above.