Enough numerical evidence prompts me to ask:

Question. Is $\sum_{k=0}^n\sum_{j=0}^k\binom{k}j^2\binom{2j}j(2j+1)^2$ divisible by $(n+1)^2$?

Convincing numerical evidence prompts me to ask:

Question. Is $\sum_{k=0}^n\sum_{j=0}^k\binom{k}j^2\binom{2j}j(2j+1)^2$ divisible by $(n+1)^2$?