Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they conjectured that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.

On the sum of parts with multiplicity at least 2 in all the partitions of n

In this paper, we investigate the sum of distinct parts that appear at least 2 times in all the partitions of [Formula: see text] providing new combinatorial interpretations for this sum. A connection with subsets of [Formula: see text] is given in this context. We provide two different proofs of our results: analytic and combinatorial. In addition, considering the multiplicity of parts in a partition, we provide a combinatorial proof of the truncated pentagonal number theorem of Andrews and Merca.

A formula for the number of partitions of n in terms of the partial Bell polynomials

We derive a formula for $p(n)$ (the number of partitions of n) in terms of the partial Bell polynomials using Faà di Bruno’s formula and Euler’s pentagonal number theorem.

A new look on the truncated pentagonal number theorem

Two new infinite families of inequalities are given in this paper for the partition function p(n), using the truncated pentagonal number theorem.