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We derive a formula for the number of non-negative integer solutions of the equation $a_1x_1 + a_2x_2 + \cdots + a_mx_m = n$ in terms of the partial Bell polynomials via the Faa di Bruno’s formula.

A formula for the number of non-negative integer solutions of a1x1 + a2x2 + ··· + amxm = n in terms of the partial Bell polynomials