-1
$\begingroup$

Consider the generating function $g(x)=\frac{(1+x)(x^2+x^3+x^4+x^5)^2}{(1-x)^4}$

Determinate the coefficient of $x^{15}$

First of all I'm very new in Discrete Mathemathics, I'm taking this subject for the first time. I got some exercises from my teacher and I'm trying to do them. I have no idea how to start on this type of problem. Any help would be great. Thanks in advance.

  • 0
    You could start by considering what are the ways to obtain $15$ summing integers. – 2017-02-21

2 Answers 2

0

HINT

If you have a sequence $A = (a_n)_{n \in \mathbb{N}}$, the function $$ G(x) = \sum_{n=0}^\infty a_n x^n $$ is called the generating function for the sequence $A$.

You are looking for the coefficient of $x^{15}$ in $g(x)$ you are given. Construct its MacLaurin series, and you will get the coefficient.

It will help to know a couple useful generating functions, e.g. $$ \frac{1}{1-x} = \sum_{n=0}^\infty x^n $$

0

Hint...You could start by extracting the factor $x^2$ and simplifying the expression, so that you are looking for the coefficient of $x^{13}$ in the expansion of $$\frac{(1+x)^3(1+x^2)^2}{(1-x)^4}$$

So now you have seven terms to find and add up...

  • 0
    How did you get to that expression? How did you extract $x^2$? – 2017-02-22
  • 0
    Yes and I simplified the larger bracketed term using the formula for the sum of a geometric series – 2017-02-22