Closed formula for the number of τ-bounded compositions

Develop a closed-form formula for |Comp(τ)|, the number of τ-bounded compositions p ∈ N^n with 0 ≤ p_j ≤ τ_j for 1 ≤ j ≤ n and Σ_{j=1}^n p_j = τ_1.

Background

Triangular weights reduce analysis to matrices M_τ indexed by τ-bounded compositions. While such counts can be approached via generating series or inclusion–exclusion, the authors remark that no closed formula for |Comp(τ)| is known, and an explicit formula would streamline spectral analysis in the triangular case.