6 problems

Let $A^{(s)}(n)=\sum_{k=0}^n\binom{n}{k}^s\binom{n+k}{k}^s$ be the generalized Franel numbers, and let $m$ be a positive integer. An Apéry limit is a limit of the quotient of two s…

Minimal-order telescoping recurrence conjecture. A telescoping recurrence equation of the conjectured minimal order $m$ exists for $s\leq 10$. Computations of Perlstadt and McIntos…

Let $A^{(s)}(n)$ be the generalized Franel numbers, let $A^{(s)}_1(n)$ denote the first auxiliary sequence arising from the expansion used to obtain the Apéry limits, and define ……

For nonnegative integers $n$ and $r$, define … In particular, $f_n^{(3)}=f_n$ are the Franel numbers. The signed Franel congruence conjecture. Let $n\geq 1$ and $r\geq 0$ be two in…

Franel-sum $3$-adic integrality conjecture. Both displayed quantities are $3$-adic integers for every positive integer $n$.

Let $p>3$ be a prime, and let $f_n=\binom{n}{0}^3+\binom{n}{1}^3+\cdots$