Unfortunately my response is probably more commiserating than helpful. I found that the resulting object from scoreDistribution is a DataDistribution object of dimension 4 with a few other named properties (although I didn't get the function above to work exactly as written):
In[34]:= n = 100;
sides = 2;
pgf = Cancel[
FactorialMomentGeneratingFunction[
DiscreteUniformDistribution[{1, sides}], z]/z]^n;
w = EmpiricalDistribution[
CoefficientList[pgf, z] -> Range[n, n sides]];
Dimensions[w]
w["Type"]
w[[1]]
w["Domain"] == w[[2]][[2]]
w["Dimension"] == w[[3]]
w["Weights"] == w[[2]][[1]]
w[[4]] (* Number of elements in the sample space *)
Out[38]= {4}
Out[39]= EmpiricalDistribution
Out[40]= "Empirical"
Out[41]= True
Out[42]= True
Out[43]= True
Out[44]= 101
I didn't figure out the associated name with w[[4]] (the number of data points).
Correction: Simpler than I thought to get the properties.
In[57]:= w["Properties"]
Out[57]= {"Dimension", "Domain", "SampleSize", "Type", "Weights"}
