We now have an expression for the polynomial formula_31 in terms of the coefficients of formula_25 (these are the formula_19s), which is what we need if we want to express a line in terms of the line above it. Recall that all the terms in a diagonal going from the upper-left to the lower-right correspond to the same power of formula_34, and that the formula_35-terms are the coefficients of the polynomial formula_25, and we are determining the coefficients of formula_31. Now, for any given formula_38, the coefficient of the formula_39 term in the polynomial formula_31 is equal to formula_41. This is indeed the simple rule for constructing Pascal's triangle row-by-row.