Let $a_n$ be the nummber of ways that 4 people can throw $n$ eyes together with a die. Every person throw once.
Now I want to find a generating function and compute $a_n$ for different $n$. To do this I first must know which recurrence relation belongs to this problem. I have no idea how to do this.
The only useful thing I have is that $4 \leq n \leq 24$.
Any help is appreciated.
On
The recursion relation is on the number of dice thrown: Let $a_{n;k}$ be the number of ways to throw a total of $n$ pips using $k$ dice. Then $$ a_{n;k} = \sum_{m=n-6}^{m-1}a_{m;k-1} $$ with initial conditions $$ a_{n;1} = \left\{ \begin{array}{cl} 1 & 1 \leq n \leq 6 \\ 0 & \text{ otherwise} \end{array} \right.$$
This recursion is easy to work with to get the number of ways for $k=4$, but it is not as easy to express $a_{n;k}$ in closed form for arbitrary $k$.
I don't think you need a recurrence relation to find the generating function: it's simply $$ \underbrace{(x+x^2+\dots+x^6)\dots(x+x^2+\dots+x^6)}_{4\text{ times}} = (x+x^2+\dots+x^6)^4.$$ Each $x^k$ term in the resulting expansion corresponds to one way to obtain $k$ from four dice.