Formulas and conjectures for partitions with restrictions on interval of parts

We focus on certain integer partitions and their weighted analogues with conditions on the interval of their parts. The q-double series turn out to be more fruitful as generating functions for our sequences. We give explicit formulas for the number of such partitions, we derive identities involving integer partitions, and we prove that some of our weighted sequences are positive. Furthermore, we state two curious conjectures on the coefficients of two q-double series.