We show that the class of Kalmar elementary functions can be inductively generated from the addition, the integer remainder and the base-two exponentiation, hence improving previous results by Marchenkov and Mazzanti. In addition, we further prove that the substitution basis defined by these three operations is minimal.

We construct arithmetic terms representing the partial sums of binomial coefficients, and we extend these results to obtain arithmetic terms representing the multisections of binomial coefficient sums. We also introduce an arithmetic term representing a certain type of multinomial coefficient sum and, as an application, we provide an arithmetic term representing the central trinomial coefficients. This solves one of the research problems of the celebrated book Concrete Mathematics, which remained open for nearly thirty years.