Number of odd binomial coefficients

Abstract

Let F ( n ) F(n) denote the number of odd numbers in the first n rows of Pascal’s triangle, and θ = ( log ⁡ 3 ) / log ⁡ 2 ) \theta = (\log 3)/\log 2) . Then α = lim sup F ( n ) / n θ = 1 \alpha = \lim \sup F(n)/{n^\theta } = 1 , and β = lim inf F ( n ) / n θ = 0.812 556 … . \beta = \lim \inf F(n)/{n^\theta } = 0.812\;556\; \ldots .