8 problems

The q-log-convexity conjecture. (i) The sequence $(\sum_{k=0}^n\binom nk^3q^k)_{n\geq8}$ is $q$-log-convex; for every integer $m\geq2$, the sequence…

Consider the triangular array with entries $\binom{n}{k}^2$ for $0\leq k\leq n$. Conditions (C1) and (C2) are the two conditions imposed on a triangular array in the source's theor…

The Narayana number $N(n,k)$ counts Dyck paths of length $2n$ with exactly $k$ peaks, and the Narayana polynomials are $N_n(q)=\sum_{k=0}^{n}N(n,k)q^k$. A sequence of polynomials i…

Let $\{f_i(q)\}_{i\geq 0}$ be any of the polynomial sequences defined in Example $\mathrm{basic\text{-}qSM}$. A sequence is infinitely $q$-log-convex if every iterate under … has c…

Let $P_n(x)$ be the Boros–Moll polynomials, whose coefficients form a polynomial sequence in $n$. A polynomial sequence is infinitely $q$-log-convex if every iterate under the oper…

Lin–Zeng's conjecture. The Jacobi–Stirling transformations of the two kinds preserve log-convexity for $z=0,1$. This conjecture concerns preservation of log-convexity by the transf…

For each $n\geq 1$, let $M_{2n}(x)=\sum_k M_{2n,k}x^k$, where $M_{2n,k}$ is the number of matchings on $[2n]$ with crossing number $k$. A polynomial sequence is infinitely q-log-co…

For each $n\geq 1$, let $P_n(x)=\sum_k P_{n,k}x^k$, where $P_{n,k}$ is the number of permutations of $[n]=\{1,2,\ldots,n\}$ whose longest increasing subsequence has length $k$. A p…