Shifted-Binomial Expansions of Dilated Binomial Polynomials: Multinomial Decimation, Reflection Symmetry, and Universal Divisibility
arXiv:2607.12173
The paper derives a closed‑form expression for the coefficients when expanding the polynomial \(\binom{pn}{p+r}\) in a shifted binomial basis, identifies a modular condition (r ≡ 1 mod p) that makes the coefficient sequence palindromic and divisible by p, and relates these sequences to known OEIS entries and Catalan numbers.
Abstract
For integers and , put and define the shifted-binomial coordinates 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 . 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 -adic valuations of and . 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 -nomial coefficient sequence.
14 pages