OEIS A000025: Partitions, ranks, and mock theta connections

We explore OEIS A000025, the sequence that records the difference between the number of partitions of n with even rank and those with odd rank. The rank is defined as the largest part minus the number of parts. For example, among the partitions of 4, the even-ranked ones are 4 and 2+2 (two of them), while the odd-ranked ones are 3+1, 2+1+1, and 1+1+1+1 (three of them), giving A000025(4) = 2 − 3 = −1. We’ll uncover how this simple rule encodes rich structure in partitions, explore its generating function and asymptotic behavior, and uncover its surprising connections to Ramanujan’s mock theta functions, hypergeometric series, and the broader world of modular forms and moonshine. We’ll also peek at how to read OEIS entries and why such a tiny sequence can illuminate deep mathematics.

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