4 problems

Let $C_n^{\vec b}$ denote the weighted Catalan number obtained by summing the weights of Dyck paths of length $2n$, where the weight sequence is $\vec b=(b_0,b_1,b_2,\ldots)$ and a…

Weighted Catalan 2-adic valuation conjecture. If $b$ satisfies

Generalized 2-adic valuation conjecture. There exists a 2-adic integer $\alpha_k$ and a nonnegative integer $c_k$ such that

Postnikov's periodicity conjecture. For $r\geq 3$, the sequence $\{L_n\pmod{3^r}\}$ is purely periodic with period $2\cdot 3^{r-3}$.