For a positive integer sequence \(\varvec{a}=(a_1, a_2,\ldots , a_{N+1})\) , Sylvester’s denumerant \(E(\varvec{a}; t)\) counts the number of nonnegative integer solutions to \(\sum _{i=1}^{N+1} a_i x_i = t\) for a nonnegative integer t. This concept has been extensively studied, and a well-known result asserts that \(E(\varvec{a}; t)\) is a quasi-polynomial in t of degree N. A milestone is Baldoni et al.’s polynomial time algorithm in 2015 for computing the top k coefficients of \(E(\varvec{a}; t)\) when k is fixed. Their development uses heavily lattice point counting theory in computational geometry. In this paper, we elucidate the contributions of Baldoni et al. within the framework of algebraic combinatorics, aiming to simplify their computation. Our research builds upon the constant term method, Barvinok’s unimodular cone decomposition, and recent results on the fast computation of generalized Todd polynomials. We introduce the CT-Knapsack algorithm, accompanied by a practical implementation in Maple, which significantly avoids repeated computations and accelerates the computation of Sylvester’s denumerant.
