Bijective Recurrences concerning Schröder paths

Consider lattice paths in Z₂ with three step types: the up diagonal (1, 1), the down diagonal (1, -1), and the double horizontal (2, 0). For n ≥ 1, let Sn denote the set of such paths running from (0, 0) to (2n, 0) and remaining strictly above the x-axis except initially and terminally. It is well known that the cardinalities, r_n = |S_n|, are the large Schröder numbers. We use lattice paths to interpret bijectively the recurrence (n+1)r_n+1 = 3(2n-1)r_n - (n-2)r_n-1, for n ≥ 2, with r₁ = 1 and r₂ = 2. We then use the bijective scheme to prove a result of Kreweras that the sum of the areas of the regions lying under the paths of Sn and above the x-axis, denoted by AS _n, satisfies AS_n+1 = 6AS_n - AS_n-1; for n ≥ 2, with AS₁ = 1, and AS₂ = 7. Hence AS _n = 1, 7, 41, 239, 1393, . . .. The bijective scheme yields analogous recurrences for elevated Catalan paths.