9 problems

Let the $n$th row of Pascal's triangle be the sequence … A sequence $(a_k)$ is log-concave if $a_k^2\geq a_{k-1}a_{k+1}$ for every relevant index $k$. A sequence is infinitely log-…

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…

Sagan's conjecture. Every row of the classical Pascal triangle $\operatorname{PD}_1$ is infinitely log-concave.

Mean-value conjecture. As $p$ tends to infinity,

Residue distribution conjecture. As the prime modulus $p$ tends to infinity,

Stephan's third ratio conjecture.

Let $C_i=\binom{n_0+id}{k_0+i\delta}$, where $n_0\geq k_0$ and $\delta>d>0$, and consider the resulting finite sequence. A finite sequence of positive numbers is Pólya frequency wh…

Let $\binom{n}{k}$ be a fixed element of Pascal's triangle, and let a ray cross a sequence of entries in the triangle. Ray-unimodality conjecture. The sequence of binomial coeffici…

For integers $n\geq0$, $u\geq0$, and $v\geq0$, consider the binomial-coefficient sequences along arithmetic lines in Pascal's triangle, such as … A sequence is infinitely log-conca…