4 problems

Let $S_k$ and $S_k^{(2)}$ be the sequences defined by the corresponding central-binomial sums in the source. Sun's integrality conjectures. For every positive integer $n$, one has…

Let $a_{2r-1}$ and $b_r$ be the integers occurring in the divisibility assertion for $S_n^{(r)}$. Guo and Liu's explicit constants conjecture. The listed values … … satisfy the pre…

Guo and Liu's existence conjecture. There exist integers $a_{2r-1}$ and $b_r$, independent of $n$, such that

Guo and Liu's congruence conjecture. One has