I'm solving a discrete math problem, but I have no idea how to solve the following problems. Regarding No. 1, should I just multiply the generating functions a (x) = 1 / 1-x and b (x) Σbn x ^ n? Would you please write what it would look like? No. 2 is b (x) = 1 / (1-x) ^ 2 as a result of differentiating a (x). Is this way of thinking correct? To be honest, I'm sorry, but I don't know what to do about numbers 3 and 4.
Suppose there are an infinite number of 1＄,5＄, and 10＄ coins. At this time, n (n = 0, 1, 2, ...) coins are allowed to be duplicated from these coins. Let fn be the total number of choices to choose. Also, let f (x) be the population function of the sequence f0, f1, f2, ....
1: Let a (x) be the generating function of the sequence 1, 1, 1, ..., and b (x) be the generating function of the sequence b0, b1, b2 .... At this time, Express the coefficient of the nth order term of a(x)b(x) using b0, b1, b2, ....
2: Let b (x) be the generating function of the sequence 1, 2, 3, ... again. Express b (x) in a closed form.
3: Explain that it is given by f (x) = (1 + x + x2 + x3 + ...) ^ 3.
4: Express fn by the formula of n.
For #1, note that \begin{align} a(x)b(x) &= \sum_{n \ge 0} x^n \sum_{k \ge 0} b_k x^k \\ &= \frac{1}{1-x}\sum_{k \ge 0} b_k x^k \\ &= \sum_{k \ge 0} b_k \frac{x^k}{1-x} \\ &= \sum_{k \ge 0} b_k \sum_{n \ge k} x^n \\ &= \sum_{n \ge 0} \left(\sum_{k=0}^n b_k\right)x^n, \end{align} so $a(x)b(x)$ is the generating function of the partial sums of $(b_n)$.
For #2, you can differentiate $\frac{1}{1-x}$, but you can also use #1 by noting that $1,2,3,\dots$ is the sequence of partial sums of $1,1,1,\dots$, so just multiply $\frac{1}{1-x}$ by $\frac{1}{1-x}$.
For #3 and #4, please start a new question.