Divisibility of binomial coefficients by powers of two

TL;DR

This paper investigates the distribution of binomial coefficients not divisible by powers of two in Pascal's triangle, showing that their counts tend to follow a normal distribution as the row number grows, using complex analytic methods.

Contribution

It establishes the normal distribution behavior of the counts of binomial coefficients not divisible by powers of two, extending previous asymptotic analyses with new complex analytic techniques.

Findings

The distribution of non-divisible entries approximates a normal distribution.

Asymptotic analysis of moments confirms the distribution pattern.

Method builds on earlier work by Drmota and colleagues.

Abstract

For nonnegative integers $j$ and $n$ let $\Theta(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are not divisible by $2^{j+1}$. In this paper we prove that the family $j\mapsto\Theta(j,n)$ usually follows a normal distribution. The method used for proving this theorem involves the computation of first and second moments of $\Theta(j,n)$, and uses asymptotic analysis of multivariate generating functions by complex analytic methods, building on earlier work by Drmota (1994) and Drmota, Kauers and Spiegelhofer (2016).