[[ Obvious growth if single symbol LHS ]]
In[]:=
Length/@MultiwayGrowthList[{"A""AAB"},{"A"},10]
Out[]=
{1,1,2,5,14,42,132,429,1430,4862,16796}
In[]:=
DiscreteAsymptoticPochhammer,-1+n,{n,Infinity,2}
-1+n
4
1
2
n!
Out[]=
-1+n
4
3
8
5/2
n
π
1
3/2
n
π
This is Catalan numbers...
In[]:=
Array[CatalanNumber,10]
Out[]=
{1,2,5,14,42,132,429,1430,4862,16796}
In[]:=
FindSequenceFunction[%]
Out[]=
CatalanNumber
FindLinearRecurrenceEquations[spec_,func_:a,var_:n]:=With[{recurrence=FindLinearRecurrence[spec]},Prepend[func[var]Total[recurrencefunc/@(var+Range[-1,-Length[recurrence],-1])]][Thread[func/@Range[Length[recurrence]]spec[[Range[Length[recurrence]]]]]]/;ListQ[recurrence]]