Divisibility of binomial coefficients by powers of two

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).

15 pages

References in corpus (2)

• Central Limit Theorem for probability measures defined by sum-of-digits function in base 2
• An explicit generating function arising in counting binomial coefficients divisible by powers of primes