Divisibility of binomial coefficients by powers of two

For nonnegative integers j and n let Θ(j,n) be the number of entries in the n-th row of Pascal's triangle that are not divisible by 2j+1. In this paper we prove that the family j↦Θ(j,n) usually follows a normal distribution. The method used for proving this theorem involves the computation of first and second moments of Θ(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).