Asymptotic estimate for the polynomial coefficients

Abstract

The polynomial coefficient $\binom {n,q}{k}$ is defined to be the coefficient of $x^{k}$ in the expansion of $(1+x+x^2+... +x^{q-1})^n$. In this note we give an asymptotic estimate for $\binom {n,q}{cn}$ as $n$ tends to infinity, where $c$ is a positive integer. Based on experimental results, it was conjectured that for any $n$, $\binom {n,q}{cn}-\binom {n,q-1}{cn}$ is unimodal and its maximum value occurs $q=\lfloor\log_{1+\frac 1{c}}{n}\rfloor$ or $q=\lfloor\log_{1+\frac 1{c}}{n}\rfloor+1$. In particular, when $c=1$, its maximum value occurs for $q=\lfloor\log_2{n}\rfloor$ or $q=\lfloor\log_2{n}\rfloor+1$.