Функции векторного разбиения в задачах суммирования для систем линейных уравнений определенного вида

We present a method for the efficient computation of weighted sums over the sets of nonneg- ative integer solutions to linear systems of a specific form. Our main result is an explicit formula that evaluates such sums through a discrete analogue of the Newton–Leibniz operator. This formula serves as a powerful tool for tackling combinatorial problems. To illustrate the effectiveness of our approach, we apply it to classical examples, including the problem of counting lucky tickets