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    Homework 5#

    Due: Thursday, October 1, 2026 23:59
    Submit: Gradescope, through Blackboard’s assignment area.

    Please write your homework by hand. Start each exercise on a new page, scan your work to a legible PDF, and clearly mark the pages associated with every exercise. For the computational parts, you may include a printout or screenshot of your Sage code and output.

    1. (A constrained sum)#

    Points: 14

    Let be the number of ordered triples of integers satisfying where is nonnegative and even, is positive and odd, and .

    1. (A) 6 points. Let , , and be the generating functions for the allowed choices of , , and , respectively. Write each generating function both as a series and in closed form.

    2. (B) 2 points. Then find the generating function for the required problem in closed form.

    3. (C) 6 points. Use Sage to find the number of solutions for , , and . Include your Sage code and output with your work (screenshot is fine)

    Need a free hint?#

    For part (C), begin with var(’x’), define using the closed form from part (A) and (B), and use A.series(variable, order) before extracting the three coefficients. Extract coefficients with A.series(variable,order).coefficient(variable,power)

    2. (Computing a closed formula for a sum of cubes)#

    Points: 16

    In this exercise we use generating functions to compute a closed formula for the sum of the cubes of numbers from to .

    Follow the steps below and show the computations used in each step.

    1. (A) 2 points. Write down the generating function in closed form for the sequence

    2. (B) 2 points. Find the generating function in closed form for the sequence

    3. (C) 2 points. Find the generating function in closed form for the sequence

    4. (D) 3 points. Find the generating function in closed form for the sum for cubes up to .

    5. (E) 6 points. Find and use that to extract the coefficient of from .

    3. (Compositions with parts 1, 2, and 3)#

    Points: 12

    Let be the number of ordered compositions of whose parts all belong to . Include one empty composition when . For example, and are different compositions of .

    1. (A) 3 points. Write the generating function for one part, and explain why counts compositions having exactly parts.

    2. (B) 4 points. Sum over all possible numbers of parts to prove that

    3. (C) 5 points. Use Sage to find . Also report the list . Include your Sage code and output with your work.

    You may use https://sagecell.sagemath.org/. The following sample shows how to extract coefficients from a power series; adapt it to this problem.

    var('x')
F = 1 / (1 - x - x^2)
T = F.taylor(x, 0, 30)
print(T.coefficient(x, 20))
print([T.coefficient(x, k) for k in range(11)])
    Need a free hint?#

    For one part, the exponent records its value. A composition has a unique number of parts, so summing the powers neither misses nor repeats any composition.

    4. (Ordered compositions versus restricted partitions)#

    Points: 10

    Let be the number of ways to write as a sum of parts chosen from where each part size may be used any number of times and order is irrelevant. Thus and represent the same choice of parts.

    1. (A) 5 points. Model with an ordinary generating function . Write and explain one factor for each allowed part size. In particular, explain why this model is a product of factors rather than the sequence construction used in Exercise 3.

    2. (B) 5 points. Use Sage to compute , , and . Include your Sage code and output with your work. Make sure your Taylor expansion includes the coefficient of .

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