8 problems

Larcombe's conjecture. Peter Larcombe conjectured that

For positive integers $r$ and $s$, define the generalized Apéry polynomials by … A polynomial sequence $(f_n(q))_{n\ge 0}$ is $q$-log-convex when the coefficients of…

The Apéry numbers are the sequences … A sequence is infinitely log-convex when every sequence obtained by iterating the log-convexity operator remains log-convex. Chen–Xia's conjec…

Let $\{P_n\}_{n\geq 0}$ be the Catalan–Larcombe–French sequence. A sequence is infinitely log-convex if every sequence obtained by repeatedly applying the operator…

Let $\mathcal{R}$ act on a sequence $\{S_n\}_{n\geq 0}$ by … Let $\{V_n\}_{n\geq 1}$ be the Fennessey-Larcombe-French sequence. Alternating $V_n$ ratio conjecture. For all integer…

Alternating ratio-sequence conjecture. For all integer $k\geq 1$, the sequence $\mathcal{R}^k(\{P_n\}_{n\geq 0})$, except for the first $k$ terms at the beginning, is log-concave i…

Let $\{V_n\}_{n\geq 0}$ be the Fennessey-Larcombe-French sequence, and consider the sequence $\{V_n^2-V_{n-1}V_{n+1}\}_{n\geq 2}$. Infinite log-convexity conjecture. The sequence…

Let $\{V_n\}_{n\geq 0}$ be the Fennessey-Larcombe-French sequence, defined by $V_0=1$, $V_1=8$, and … A sequence $\{a_n\}$ is log-concave when $a_n^2\geq a_{n-1}a_{n+1}$ for every…