I am interested in the serie $$\sum_{n=1}^\infty(2n+1)e^{-tn(n+1)}$$ for something I am working on. Wolfram alpha gets stuck on this problem, but it gives the following result for a closely related summation: $$\sum_{n=1}^\infty e^{-tn(n+1)} = \frac{\vartheta_2(0,e^{-t})}{2e^{-\frac{t}{4}}}$$ where $\vartheta_2$ is a Jacobi theta function. I was hoping anyone here would have some idea about how to proceed for this problem, or, even better, a solution.
Thank you in advance!