Mathematical model for dice

Here is a single die with six sides
In[]:=
s=6;(*sidesonadie*)​​d=
s
∑
i=1
i
x
(*modelofadie*)
Out[]=
x+
2
x
+
3
x
+
4
x
+
5
x
+
6
x
And here is the result of rolling a set of three such dice.
In[]:=
n=3;(*numberofdice*)​​r=
n
d
;(*diceroll*)​​Expand[r]
Out[]=
3
x
+3
4
x
+6
5
x
+10
6
x
+15
7
x
+21
8
x
+25
9
x
+27
10
x
+27
11
x
+25
12
x
+21
13
x
+15
14
x
+10
15
x
+6
16
x
+3
17
x
+
18
x
The exponent in each term is the sum of the dice, and the coefficient is the chance weight of that outcome.
In[]:=
c=CoefficientList[r,x];(*extractlistofchanceweights*)​​c=Drop[c,n](*trimthelisttostartatcountofdice*)
Out[]=
{1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1}
In[]:=
ListPlot[c]
Out[]=