I'm trying to understand the behavior of the coefficients of binomials, as seen in Pascal's Triangle. Here's are some examples: (x + 1)^2 = x^2 + 2x + 1; COEFFICIENTS: 1,2,1 (x + 1)^3 = x^3 + 3x^2 + 3x + 1; COEFFICIENTS: 1,3,3,1 (x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1; COEFFICIENTS: 1,4,6,4,1 As you can see, as the exponent grows, the numbers begin to take the shape where they rise and fall, like a curve (as seen in the project). If all of these coefficients are turned into their logarithms, they take a shape very similar to a parabola, so I tested whether the shape is in fact a parabola, the white curve in the picture. It is fairly clear that the shape (orange) does not match that of a parabola. I then compared the curve to that of a catenary (a more complex shape, in blue), and found that it was extremely similar, but I'm not quite sure I found a match. [Catenary formula: y = a * cosh(x/a)] The curvature of these curves do not change with different coefficients.
If anyone has any insight as to why these curves don't match (read the Instructions section), please let me know in the comments. I know Scratch is hardly the place for fancy math, but maybe someone knows what's going on!
