Let Wd(n) be the number of 2n-step walks in ℤd which begin and end at the origin. We study the exponent of 2 in the prime factorisation of this number; i.e., wd(n)=ν2(Wd(n)). We show that, for each d, there is a relationship between wd(n) and the number s2(n) of 1s in the binary expansion of n. For example, wd(n)=s2(n) if d is odd and wd(n)=2s2(n) if ν2(d)=1; while wd(n)≥3s2(n) if ν2(d)=2. The pattern changes further when ν2(d)≥3. However, for each d, we give the best analogous estimate of wd(n) together with a description of all n where equality is attained. The methods we develop apply to a wider range of problems as well, and so might be of independent interest.
