Title: A Simple Formula for Binomial Coefficients Revealed Through Polynomial Encoding

Abstract: We revisit an unconventional formula for binomial coefficients by applying an underexplored property of polynomial encoding. The formula, $\binom{n}{k} = \left\lfloor\frac{(1 + 2^{n})^{n}}{2^{n k}}\right\rfloor \bmod{2^{n}}$, is valid for $n > 0$ and $0 \leq k \leq n$. We relate this formula to existing mathematical methods via Kronecker substitution. Additionally, we generalize this formula to compute multinomial coefficients. A baseline computational complexity analysis identifies opportunities for optimization. We conclude by positing an open problem concerning the efficient computation of $\binom{n}{k}$ modulo $n$ using the formula.