What happens when row n of the Pascal triangle is shifted by k then added to itself? Or in continuous terms, two binomial distributions, identical apart from the shift along the x-axis, are summed? Consider k fixed and n increasing: initially n_C_i + n_C_j where i+j = n+k is bimodal; but at some point n_0 it becomes unimodal and remains so for n > n_0. Numerical experiments suggest that n_0 ~ k^2 - 5/3 ; indeed this bound estimate is remarkably good, being only <0.1 too large at k = 2 , and <0.001 too large at k = 20 . No doubt probability wonks know all about this stuff already: so can anyone point me at a reference to a proof of the bound; or alternatively explain why it's perfectly obvious? Fred Lunnon