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 $(n+1)^2$?
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Divisibility of a binomial sequence
The answer is yes, and the proof can be found in V. J. W. Guo, J.-C. Liu, Proof of some conjectures of Z.-W. Sun on the divisibility of certain double sums, Int. J. Number Theory 12 (2016), 615-623. An arXiv version is also [available](https://arxiv.org/abs/1412.5415).


