Formulas for the number of binomial coefficients divisible by a fixed power of a prime

Abstract

Define ${\theta _j}(n)$ as the number of binomial coefficients $\binom {n}{s}$ divisible by exactly ${p^j}$. A formula for ${\theta _2}(n)$ is found, for all $n$, and formulas for ${\theta _j}(n)$ for $n = a{p^k} + b{p^r}$ and $n = {c_1}{p^{{k_1}}} + \cdots + {c_m}{p^{{k_m}}}$ (${k_1} \geqq j$, ${k_{i + 1}} - {k_i} \geqq j$ for $i = 1$, …, $m - 1$) are derived.

Locations

• Proceedings of the American Mathematical Society - View - PDF