2 problems

For $r\geq 1$ and $0\leq a\leq r$, consider all indices $(k_1,\ldots,k_r)$ with $k_i\in\{1,2\}$ and exactly $a$ entries equal to $2$. Let $\#\{i\mid i\text{ odd},\ k_i=2\}$ count t…

Weighted sum formula conjecture. For every positive integer $n\ge m$, there exist polynomials $e_{F,k}(x),c_{F,k}(x)\in\Q[x]$, depending only on $k$ and $F$, such that