LDU decomposition of an extension matrix of the Pascal matrix

Let Ω_m,n(α,β,γ) denote a set of all elements of weighted lattice paths with weight (α,β,γ) in the xy-plane from (0,0) to (m,n) such that a vertical step V=(0,1), a horizontal step H=(1,0), and a diagonal step D=(1,1) are endowed with weights α,β, and γ respectively and let ω(Ω_m,n(α,β,γ)) denote the weight of Ω_m,n(α,β,γ) defined by