Shifted-Binomial Expansions of Dilated Binomial Polynomials: Multinomial Decimation, Reflection Symmetry, and Universal Divisibility

Abdelhai Doukali

For integers $m\ge 2$ and $r\ge 1$, put $d=m+r-1$ and define the shifted-binomial coordinates $B_{m,r,k}$ by \[ \binom{mn}{d} = \sum_{k=1}^{d+1}B_{m,r,k}\binom{n+k-1}{d}. \] The main…