a(n)=Number of Unique Matrix Products in (A+B+C)^n When [A,B]=0

[Paul D. Hanna]

Seqfans,
      Max's nice work has provided some insights into the combinatorics
behind my queries. 
Below are formulas for 2 of the sequences that seem to lend themselves
readily to generalization. 
   
> > a(n) = the number of unique products of matrices A, B, C, in
(A+B+C)^n 
> > where commutator [A,B]= 0 but C does not commute with A or B. 
> It is in the OEIS under the name A001906  (Max)
FORMULA: 
a(n) = Sum_{i=0..n} Sum_{j=0..n} C(n-j,i)*C(n-i,j) 
  
> > "a(n) = number of unique matrix products in (A+B+C+D)^n  where 
> > commutator [A,B]=[A,C]=[B,C]=0 but D does not commute with A, B, or
C." 
> This one seems to be A052529  (Max)
FORMULA: 
a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n}
C(n-j-k,i)*C(n-i-k,j)*C(n-i-j,k)
 
 From these (assuming true), can anyone find a more general formula? 
How about formulas for these sequences (given earlier): 
 
(A1) "a(n) = number of unique matrix products in (A+B+C+D)^n  where 
commutator [A,B]=0 but neither A nor B commutes with C or D."  
  
(A2) "a(n) = number of unique matrix products in (A+B+C+D)^n  where 
commutators [A,B]=[C,D]=0 but neither A nor B commutes with C or D."  
 
   
I would like to take a wild guess at the formula for this case: 
 
(A4) "a(n) = number of unique matrix products in (A+B+C+D+E)^n  
where A,B,C, and D all commute with each other, but not with E." 
 
FORMULA (predicted from trend): 
a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{m=0..n} 
          C(n-j-k-m,i)*C(n-i-k-m,j)*C(n-i-j-m,k)*C(n-i-j-k,m) 
which would begin:
   1,5,19,69,250,907,3292,11949,43371,157422,571388,...
which is OEIS: A055991  (a(n) is its own 4th difference).   
 
Would someone like to enumerate (A4) to see if it matches the formula?
  
Thanks,
      Paul
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