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

Abstract

For integers $m\ge2$ and $r\ge1$, 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 purpose of this paper is to identify these coordinates with a specific residue-class decimation of a multinomial coefficient sequence. We prove \[ B_{m,r,k} = [x^{mk-1}](1+x+\cdots+x^{m-1})^{m+r}. \] This identity converts the alternating finite-difference formula for the coordinates into a positive coefficient formula, and yields their exact support. We then prove that the nonzero coefficient vector is palindromic if and only if $r\equiv2\pmod m$. This symmetry criterion is derived here directly within the multinomial-decimation framework. The principal arithmetic results are an unconditional divisibility law and an exact formula for the greatest common divisor of each nonzero row. We prove \[ \frac{m}{\gcd(m,r-1)}\mid B_{m,r,k}, \] and, more precisely, determine \[ \gcd_{1\le k\le K_{m,r}} B_{m,r,k} \] as an explicit product of prime powers determined by the $p$-adic valuations of $m$ and $m+r-1$. Finally, a roots-of-unity filter gives the exact row sum \[ \sum_k B_{m,r,k}=m^{m+r-1}. \] Thus the shifted-binomial coordinates form a positive, arithmetically structured decimation of an $m$-nomial coefficient sequence.