The p-adic valuation of k-central binomial coefficients

TL;DR

This paper investigates the divisibility properties of a family of coefficients c(n,k), generalizing central binomial coefficients, proving their integrality and exploring their p-adic valuations.

Contribution

It establishes the integrality of c(n,k) and analyzes their p-adic valuations, extending understanding of binomial coefficient generalizations.

Findings

c(n,k) are integers for all n,k

Derived formulas for p-adic valuations of c(n,k)

Connected c(n,k) to classical binomial coefficients for k=2

Abstract

The coefficients c(n,k) defined by (1-k^2x)^(-1/k) = sum c(n,k) x^n reduce to the central binomial coefficients for k=2. Motivated by a question of H. Montgomery and H. Shapiro for the case k=3, we prove that c(n,k) are integers and study their divisibility properties.