Let $a_n$ be the number of ways in which the score $n$ can be obtained by throwing a die any number of times. Show that the generating function of $\{a_n\}$ is $(1-s-s^2-s^3-s^4-s^5-s^6)^{-1} -1$.
I think that the generating function of $\{a_n\}$ is $$\frac1{(1-s)(1-s^2)(1-s^3)(1-s^4)(1-s^5)(1-s^6)}.$$
The $n$ th coefficient of this generating function represent the number of ways $a_n$. But, how can this equation be be reduced to $(1-s-s^2-s^3-s^4-s^5-s^6)^{-1} -1$?
I would appreciate if you give some help.
Those aren't the same. The coefficient of $s^3$ in $(1-s-s^2-\cdots-s^6)^{-1}-1$ is $4$ which corresponds to the sequences of dice throws $(1,1,1)$, $(1,2)$, $(2,1)$ and $(3)$.
But the coefficient of $s^3$ in $1/[(1-s)(1-s^2)\cdots(1-s^6)]$ is $3$. In effect, this generating function is counting both $(1,2)$ and $(2,1)$ just as one outcome. The first generating function counts ordered sequences of dice throws adding to $n$, but the second generating function counts unordered sequences of dice throws adding to $n$.