arithmetic significance of coefficients in a power series

Question

In this answer user @JackD'Aurizio said

$$\frac{1}{1-x^3-x^4-x^{20}}=\sum_{k\geq 0}(x^3+x^4+x^{20})^k $$ and the coefficient of $x^{46}$ in $(x^3+x^4+x^{20})^k$ is the cardinality of the $k$-tuples with coordinates in $\{3,4,20\}$ such that the sum of the coordinates equals $46$.

and that made me really intrigued. Previously, I had only considered the significance of the coefficients of such a power series to be that they satisfied a certain recurrence relation. This new way of thinking about the coefficients if much more number-theoretic and interesting.

Certainly there is a more general theory behind this line of thought. What is it? Could I have some links where I could learn more? Thanks.

Answer 1

Polya, Szego - Problems and theorems in Analysis I, part one, chapter $1$ - operations with power series.

Answer 2

I agree, it's a radical way to think about numbers and functions! It's called a Generating Function. Many resources related to generating functions can be found in the answers to this question: How can I learn about generating functions?

At the risk of being repetitious, here is an anecdote about the statistician Frederick Mosteller's first encounter with generating functions.

A key moment in my life occurred in one of those classes during my sophomore year. We had the question: When three dice are rolled what is the chance that the sum of the faces will be 10? The students in this course were very good, but we all got the answer largely by counting on our fingers. When we came to class, I said to the teacher, "That's all very well - we got the answer - but if we had been asked about six dice and the probability of getting 18, we would still be home counting. How do you do problems like that?" He said, "I don't know, but I know a man who probably does and I'll ask him." One day I was in the library and Professor Edwin G Olds of the Mathematics Department came in. He shouted at me, "I hear you're interested in the three dice problem." He had a huge voice, and you know how libraries are. I was embarrassed. "Well, come and see me," he said, and I'll show you about it." "Sure, " I said. But I was saying to myself, "I'll never go." Then he said, "What are you doing?" I showed him. "That's nothing important," he said. "Let's go now."

So we went to his office, and he showed me a generating function. It was the most marvelous thing I had ever seen in mathematics. It used mathematics that, up to that time, in my heart of hearts, I had thought was something that mathematicians just did to create homework problems for innocent students in high school and college. I don't know where I had got ideas like that about various parts of mathematics. Anyway, I was stunned when I saw how Olds used this mathematics that I hadn't believed in. He used it in such an unusually outrageous way. It was a total retranslation of the meaning of the numbers. [Albers, More Mathematical People].