Three recurrences for parallelogram polyominoes

We define three polynomials sequences , for α=0, 1, or 2, in terms of various "diagonal thickness" parameters on parallelogram polyominoes. Here n denotes the semiperimeter and w marks th horizontal width of a polyomino. We extend a rcent bijective proof of Foata and Zeilberger and a recent generating function proof of Rogers, Shapiro, and Woan to obtain recurrences for these sequences of the form where and α=0, 1, or 2. Concomitantly, we find that the polynomial sequences generalize some well-known sequences: for instance are the Catalan numbers are the Schroder numbers are the square-triangular numbers, and are the central binomial coefficients.