## Adjustable Pascal's Triangle
The table in this initial position is the form in which Pascal wrote his "Arithmetical Triangle" (his name for it.) John Wallis also used this form, and interpolated values for the halfway in-between rows and columns. (See the WallisTable applet.) Isaac Newton realized that if you change the shape so that the column of 1s goes down and to the right at a 45 degree angle, then the entries become the coefficients in a power series. (See the Lecture Notes on Newton's Binomial Series.)
You can also put this in the traditional modern form by making an isosceles triangle with A at the apex. Try other positions, and see what patterns you can find.
Susan Addington, Created with GeoGebra |