Taking a short cut through Pascal's triangle

摘要

Everybody knows the proper way to travel through Pascal's triangle, starting at the top and moving down either left (L) or right (R), with each number in the triangle giving the total number of ways of reaching that position. But why restrict ourselves to left and right moves only, zigzagging our way down the triangle, when we could be going much more directly? We are crying out for vertical moves downwards to the next number below (that is, two rows lower). Let us call this sort of move D and consider the effect of having L, R and D on the number of possible ways of reaching each element. It somehow seems fitting that in a triangle there should be three possible moves. Putting a 1 at the top and continuing down gives us what I will call the short-cut triangle, the first few rows of which are shown Figure 1.