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A truncated theta-type series with triangular weights

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.CO — Combinatorics

Contributed by Manjil Saikia ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

Merca (2026) recently introduced the two-block odd partition function a(n)a(n), proved that its generating function factors as A(q)=P(q)S(q)A(q)=P(q)S(q), where P(q)P(q) is the generating function of partitions whose odd parts are distinct and whose even parts are unrestricted and S(q)S(q) is a theta-type series with triangular-number weights, and conjectured that for N≡1,2(mod4)N\equiv1,2\pmod 4 the truncation P(q)SN(q)P(q)S_N(q) of this product has nonnegative coefficients, positive from q3q^{3} onward. We prove the conjecture in the stronger form $P(q)S_N(q)-A(q)\in\NN[[q]]$.

Provenance statement

The author used Anthropic's Claude Fable 5.1 to generate the skeleton of the arguments presented here (based on the ideas and proof strategy used in the author's recent AI-free co-authored paper [SS26]). After several rounds of edits, the final form of the proofs took shape. We have followed closely the strategy used in [SS26]. The final version of the manuscript was reviewed using OpenAI's GPT-6 Sol for any errors and redundancies.

Tools used

Anthropic
Claude FableVersion 5.1
OpenAI
ChatGPTVersion 6 Sol
Wolfram
Mathematica

References

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  7. G. Pólya and G. Szegő, Problems and Theorems in Analysis. I , Classics in Mathematics, Springer-Verlag, Berlin, 1998.
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Version history

  1. v1Submitted by Manjil SaikiaInitial depositCurrentOct 22, 2026