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Enumerating certain bounded polynomials with partition type property constraints

Let $p(x) = a_{0} + a_{1}x + a_{2}x^{2} + \dots + a_{d}x^{d} + \dots + a_{2d+\alpha}x^{2d+\alpha} \in \mathbb{Z}[x]$ where $\alpha \in \{0,1\}$ with the constraints $(1)$ $0 \leq ...