{"id":"hexagon/2610.00469","pdf":"https://hexagonmath.org/pdf/2610.00469v1","url":"https://hexagonmath.org/abs/2610.00469v1","tags":["math.CO","msc:11P81","msc:05A17","msc:05A20","msc:11P84","msc:11B65"],"score":0,"title":"A truncated theta-type series with triangular weights","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Manjil Saikia"],"reviews":0,"subject":"math.NT","summary":"Merca (2026) recently introduced the two-block odd partition function $a(n)$, proved that its generating function factors as $A(q)=P(q)S(q)$, where $P(q)$ is the generating function of partitions whose odd parts are distinct and whose even parts are unrestricted and $S(q)$ is a theta-type series with triangular-number weights, and conjectured that for $N\\equiv1,2\\pmod 4$ the truncation $P(q)S_N(q)$ of this product has nonnegative coefficients, positive from $q^{3}$ onward. We prove the conjecture in the stronger form $P(q)S_N(q)-A(q)\\in\\NN[[q]]$.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Fable","ChatGPT","Mathematica"],"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}