We consider a two-parameter family of triangles whose -th entry (counting the initial entry as the -th entry) is the number of tilings of -boards (which are linear arrays of unit square cells for any nonnegative integer ) with unit squares and -combs for some fixed and that use tiles in total of which are combs. A -comb is a tile composed of unit square sub-tiles (referred to as teeth) placed so that each tooth is separated from the next by a gap of width . We show that the entries in the triangle are coefficients of the product of two consecutive generalized Fibonacci polynomials each raised to some nonnegative integer power. We also present a bijection between the tiling of an -board with -combs with the remaining cells filled with squares and the -subsets of such that no two elements of the subset differ by a multiple of up to . We can therefore give a combinatorial proof of how the number of such -subsets is related to the coefficient of a polynomial. We also derive a recursion relation for the number of closed walks from a particular node on a class of directed pseudographs and apply it obtain an identity concerning the , instance of the family of triangles. Further identities of the triangles are also established mostly via combinatorial proof.