Let $W_{s,r,n}$ be the total number of ways that the sum $s$ can be displayed after throwing $r$ number of $n$-sided dice. Define
$$W_{s,0,n} = \begin{cases} 1, & \text{if s = 0} \\ 0, & \text{if s $\neq$ 0} \\ \end{cases} $$
for all $s \in \mathbb Z$ and $n \in \mathbb N^*$. Can you prove that
$$W_{s,r,n} =W_{s-1,r,n} + W_{s-1,r-1,n} - W_{s-(1+n),r-1,n}$$
for all $s \in \mathbb Z$, $r \in \mathbb N^*$, and $n \in \mathbb N^*$?
You regard the dice as distinguishable, so for example $1+5+6+6$ is distinct from $6+5+1+6$. So $\displaystyle W_{s,r,n} = \sum_{j=1}^n W_{s-j,r-1,n}$.
There is a proof by induction:
Base case: for $r=1$ the recurrence is satisfied by $$W_{s,1,n} = \begin{cases} 1, & \text{ if } 1 \le s \le n \\ 0, & \text{ if } s \le 0 \text{ or } s \ge n+1 \\ \end{cases}$$
Inductive case: if it is true for up to $r-1$ dice then
$$W_{s,r,n} = \sum_{j=1}^n W_{s-j,r-1,n} = \sum_{j=1}^n \left(W_{s-j-1,r-1,n} + W_{s-j-1,r-2,n} - W_{s-j-(1+n),r-2,n}\right)$$ $$= \sum_{j=1}^n W_{s-j-1,r-1,n} + \sum_{j=1}^n W_{s-j-1,r-2,n} - \sum_{j=1}^n W_{s-j-(1+n),r-2,n} $$ $$= W_{s-1,r,n} + W_{s-1,r-1,n} - W_{s-(1+n),r-1,n}$$
so it is true for up to $r$ dice, and so for any positive number of dice.