2 problems

Terminal-block symmetric sum conjecture. Let $n,j_0,\ldots,j_{2n}$ be non-negative integers and put $m=j_0+\cdots+j_{2n}$. Then

Symmetric sum conjecture. Let $n$ be a positive integer, let $j_0,\ldots,j_{2n-1}$ be non-negative integers, and put $m=j_0+\cdots+j_{2n-1}$. Then