On some conjectural supercongruences involving the sequence $t_n(x)$

Hui-Li Han, Chen Wang

In this paper, we study some supercongruences involving the sequence $$t_n(x)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{x}{k}\right)\left(\genfrac{}{}{0ex}{}{x+k}{k}\right)2^k$$ and solve some open problems. For any odd pr…