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. 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. This is explained in the book Connecting History to Secondary School Mathematics by Carrejo, Dennis, and Addington, to be published by Springer Verlag in 2025.
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.