math.NT — Number Theory

A truncated theta-type series with triangular weights

Contributed by Manjil Saikia

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 $P(q)S_N(q)-A(q)\in\NN[[q]]$.