7 problems

Sun's ratio monotonicity conjecture. The sequence $\{S_{n+1}/S_n\}_{n\geq 3}$ is strictly increasing to the limit $9$, and the sequence…

For $k\in\mathbb{N}$, consider the Pascal-triangle column sequence $\{\binom{n+k}{k}\}_{n=0}^{\infty}$. McNamara–Sagan conjecture. This sequence is infinitely log-concave, meaning…

Amdeberhan–Moll–Vignat conjecture. Both sequences $\{a_n\}_{n\ge 1}$ and $\{b_n\}_{n\ge 1}$ are log-convex. The conjecture concerns recursively defined integer sequences arising al…

Generalized Glasby–Paseman sequence conjecture. This sequence satisfies the following properties:

Sun's conjecture. For every positive integer $n$,

For $x>0$, define … so that $B(n)=B_n$ for nonnegative integers $n$, where $B_n$ is the $n$-th Bell number. Bell-function concavity conjecture. The function $B(x)^{1/x}$ is log-con…

Let A=a_i_{0\leq i\leq\infty} be a sequence, and define … A sequence is $k$-log-convex if $\mathcal{L}^j(A)$ is log-convex for $j=0,1,\ldots,k-1$, and infinitely log-convex if…