A polynomial time algorithm for Sylvester waves when entries are bounded

The Sylvester's denumerant $d(t; \boldsymbol{a})$ is a quantity that counts the number of nonnegative integer solutions to the equation $\sum_{i=1}^{N} a_i x_i = t$, where $\boldsymbol{a} = (a_1, \dots, a_N)$ is a sequence of distinct positive integers with $\gcd(\boldsymbol{a}) = 1$. We present a polynomial time algorithm in $N$ for computing $d(t; \boldsymbol{a})$ when $\boldsymbol{a}$ is bounded and $t$ is a parameter. The proposed algorithm is rooted in the use of cyclotomic polynomials and builds upon recent results by Xin-Zhang-Zhang on the efficient computation of generalized Todd polynomials. The algorithm has been implemented in \texttt{Maple} under the name \texttt{Cyc-Denum} and demonstrates superior performance when $a_i \leq 500$ compared to Sills-Zeilberger's \texttt{Maple} package \texttt{PARTITIONS}.