Edric Cane came up with a simple way to establish any row in Pascal’s triangle, creating a simple sequence of fractions that, when multiplied successively, will produce the numbers in any desired row. Here’s an example for Row 7, giving the coefficients for (a + b)7 = a7 + 7a6b + 21a5b2 + 35a4b3 etc.:
Another example, for Row 10:
The same can be done for any desired row.
from Futility Closet https://www.futilitycloset.com/2018/01/13/a-triangle-calculator/