Putnam 1978 B2

Express $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^2 n +m n^2 +2mn }$ as a rational number.

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.