Number of binomial coefficients divided by a fixed power of a prime

TL;DR

This paper presents a general formula for counting binomial coefficients that are divisible by a fixed power of a prime number, providing a precise enumeration based on prime power divisibility.

Contribution

It introduces a novel, general formula for counting binomial coefficients divisible by a specific prime power, extending previous understanding of binomial coefficient divisibility.

Findings

Provides a formula for the count of binomial coefficients divisible by p^j

Enables precise enumeration of binomial coefficients with specific prime power divisibility

Extends classical results on binomial coefficient divisibility

Abstract

We state a general formula for the number of binomial coefficients $n$ choose $k$ that are divided by a fixed power of a prime $p$, i.e., the number of binomial coefficients divided by $p^j$ and not divided by $p^{j+1}$.