Pascal’s Triangle was developed by a French mathematician named Blaise Pascal. The rows are numbered starting with 0 and go all the way to n, just like list indices in Python. The triangle always starts with a single term in the 0th row, which is a [1]. The nth row of the triangle has n+1 terms, and each term is found by adding the values of the two terms above it. For example, row number 2 (actually the third row) has three terms: [1, 2, 1]. The row above it, row number 1, is [1, 1], and the row above that one, row 0, is just [1]. Pascal’s Triangle has several uses in mathematics, such as binomial expansions. (http://en.wikipedia.org/wiki/Pascal%27s_triangle contains more information.) See how the terms are obtained .
Task 1:
Your first task is to create a function, pascal_row, that outputs a list of elements found in a certain row of Pascal’s Triangle. The pascal_row function should take a single integer as input, which represents the row number. Row number 0 indicates the single term row, [1]. The input to the pascal_row function will always be an integer greater than or equal to 0. Hopefully you have noticed that each row in Pascal’s Triangle can be computed by recursively calculating the row above it. Be sure to think carefully about the base case in this problem. Also, it may be helpful to write a helper function that creates a new list of sums of adjacent terms in the original list.
Here are some examples of calling the pascal_row function:
>>> pascal_row(0) [1]
>>> pascal_row(1) [1, 1]
>>> pascal_row(5)
[1, 5, 10, 10, 5, 1]
Task 2:
Write a second function called pascal_triangle that takes as input a single integer n and returns a list of lists containing the values of the all the rows up to and including row n.
Here are some examples of calling the pascal_triangle function:
>>> pascal_triangle(0)
[[1]]
>>> pascal_triangle(1) [[1], [1, 1]]
>>> pascal_triangle(5)
[[1], [1, 1], [1, 2, 1], [1, 3, 3, 1], [1, 4, 6, 4, 1], [1, 5, 10, 10, 5, 1]]
Task 3:
Write two functions, named test_pascal_row and test_pascal_triangle,with no
arguments. They should do what the name suggests – test the corresponding function. There should be at least four tests each. For testing, use assertions, in the way shown in the file assert_LCS_change.py that was posted online. In brief, each test asserts that a call of your function equals the value expected.
