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Interpreting the Truncated Pentagonal Number Theorem using Partition Pairs

Interpreting the Truncated Pentagonal Number Theorem using Partition Pairs

In 2012 Andrews and Merca gave a new expansion for partial sums of Euler's pentagonal number series and expressed \[\sum_{j=0}^{k-1}(-1)^j(p(n-j(3j+1)/2)-p(n-j(3j+5)/2-1))=(-1)^{k-1}M_k(n)\] where $M_k(n)$ is the number of partitions of $n$ where $k$ is the least integer that does not occur as a part and there are more parts greater than $k$ …